Emptiness Formation Probability and Quantum Knizhnik-Zamolodchikov Equation

H. E. Boos ^{1}

Max-Planck Institut für Mathematik Vivatsgasse 7, 53111 Bonn, Germany

V. E. Korepin C.N. Yang Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook, NY 11794–3840, USA

F.A. Smirnov ^{2} LPTHE, Tour 16, 1-er étage, 4, pl. Jussieu 75252, Paris Cedex 05, France

Abstract

We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of formation of a ferromagnetic string P (n) in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of the EFP in the inhomogeneous case. It is based on the quantum Knizhnik-Zamolodchikov equation [qKZ]. We calculate EFP for n ≤ 6 for inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations and number theory. We also make a conjecture about a structure of EFP for arbitrary n.

1on leave of absence from the Institute for High Energy Physics, Protvino, 142284, Russia 2Membre du CNRS

1. Introduction

The Hamiltonian of the XXX Heisenberg spin chain can be written like this

H = ∑_{i=−N}^{N} (σ_{i}^{x}σ_{i+1}^{x} + σ_{i}^{y}σ_{i+1}^{y} + σ_{i}^{z}σ_{i+1}^{z} − 1 )  (1.1)

Here 2N +1 is the length of the lattice and σ_{i}^{x}, σ_{i}^{y}, σ_{i}^{z} are Pauli matrices. We consider thermodynamic limit [ N goes to infinity] . The sign in front of the Hamiltonian indicates that we are dealing with the antiferromagnetic case. We also imply periodic boundary conditions. The model was solved by Bethe in 1931, see [1]. The ground state was constructed by by Hulthén, see [2]. We shall denote the ground state in the thermodynamic limit by |GS⟩ . The emptiness formation probability (EFP) for the XXX model is defined as follows:

P (n) = ⟨GS| ∏_{j=1}^{n} P_{j}|GS⟩,  (1.2)

here P_{j} = S_{j}^{z} + \frac{1}{2} is the projector on the state with the spin up in the j-th lattice site. The integer n has a meaning of a length of a continuous ferromagnetic string. P (n) is a probability that this string can appear in the antiferromagnetic ground-state. The importance of EFP was emphasized in [3].

P (n) can be represented as n-multiple integral. The integral representation follows from the work of RIMS group. RIMS approach is based on vertex operators and bosonic representation of infinite-dimensional quantum algebras, see [4, 5]. The explicit formula for P (n) in XXX limit was obtained in [7]:

P (n) = ∏_{j=1}^{n} ∫_{C} \frac{dλ_{j}}{2πi} U_{n}(λ_{1}, …, λ_{n}) T_{n}(λ_{1}, …, λ_{n})  (1.3)

where

U_{n}(λ_{1}, …, λ_{n}) = π^{\frac{n(n+1)}{2}} ∏_{1≤k<j≤n} sinh π(λ_{j} − λ_{k}) ; ∏_{j=1}^{n} sinh^{n} πλ_{j}  (1.4)

and

T_{n}(λ_{1}, …, λ_{n}) = \frac{∏_{j=1}^{n} λ_{j}^{j−1}(λ_{j} + i)^{n−j}}{∏_{1≤k<j≤n}(λ_{j} − λ_{k} − i)}  (1.5)

The contour C goes parallel to the real axis with the imaginary part confined between 0 and −i for each integral.

In papers [8, 9, 10] we evaluated the integrals for n ≤ 5. We discovered that these P (n) can be expressed in terms of the values of Riemann zeta function at odd arguments, log 2 and rational coefficients. We conjectured that this is a general property for all P (n). In this paper we proved the property for P (6). We think that all correlation functions

⟨GS|σ_{i_{1}}^{z} σ_{i_{2}}^{z} … σ_{i_{m}}^{z}|GS⟩  (1.6)

also have this property. Asymptotic behavior of P (n) for large n was studied in the papers [8],[10],

[13] and [14]. The technique of calculation of these integrals, described in the paper [9] worked for n = 2, 3, 4. In the paper [10] we calculated P (5) by means of this technique. However, these computations are so complicated that it is problematic to generalize them to the case n = 6. So we start looking for indirect methods of evaluation of the integrals (1.3).

It appeared to be useful to consider inhomogeneous case. In this case there are more free parameters. We call them inhomogeneity parameters and denote by z_{1}, …, z_{n}. The EFP in the inhomogeneous case we shall denote by P_{n}(z_{1}, …, z_{n}). Let us remind to the reader that inhomogeneous models were used for evaluation of correlation functions from the very beginning. For the massive regime of the XXZ model the vertex operator approach was developed in [4, 5]. It allowed to express the correlation functions in terms of the trace functions. Special combinations of these trace functions satisfy the quantum Knizhnik-Zamolodchikov equation (qKZ), see [11, 12]. Later in paper [6] Miwa and Jimbo suggested that the correlation functions in the gap-less regime directly satisfy the qKZ equation. Since the XXX model belongs to the gap-less regime we shall use qKZ for evaluation of EFP in the inhomogeneous case. We suggest a general ansatz for P_{n}(z_{1}, …, z_{n}), see (3.20). This constitute a new method for computation of the EFP. On the other hand, it is easy to generalize the technique explained in [9] to the inhomogeneous case and calculate the EFP directly [for short ferromagnetic strings]. When we can compare results, they coincide.

The paper is organized as follows. In the Section 2 we discuss the relation of EFP to the qKZ and derive three important properties of the EFP in the inhomogeneous case. In Section 3 we apply a generalization of the technique described in [9] to the inhomogeneous case and compute P_{n} directly for n ≤ 4. Then we check that these P_{n} satisfy all the properties, which follow from qKZ. This helps us to formulate a general ansatz for P_{n} in the inhomogeneous case. Further, we suggest a new way of computing P_{n}. One can use the ansatz and general properties of P_{n} [which follow from the qKZ]. In this way we get the explicit expressions for P_{5} and P_{6} in the inhomogeneous case. In Section 4 we discuss the homogeneous limit of P (n) for n ≤ 6. In particular, when n ≤ 5 we reproduce our previous results, obtained in [8, 9, 10]. We also get the analytic expression for P (6) in the homogeneous limit. Having this answer we can compare it with the numerical value for P (6) obtained by the DMRG method in [10]. We also discuss the structure of EFP in the homogeneous case and offer some plausible conjectures. In the last Section 6 we discuss the results and outline some possible ways of a further progress.

2. The EFP in the inhomogeneous case and the qKZ equation.

We believe that consideration of the inhomogeneous case instead of the homogeneous one can give us a new information about the EFP and other correlation functions.

A method of calculation of correlation functions, which we use was found in the papers [4, 5]. It is based on theory of infinite-dimensional quantized algebras and vertex operators. We shall need elements of this method. Let us introduce some notations.

We use the R-matrix:

R(λ) = \frac{R_{0}(λ)}{λ + πi} ( \begin{matrix}λ + πi & 0 & 0 & 0 \\ 0 & −λ & πi & 0 \\ 0 & πi & −λ & 0 \\ 0 & 0 & 0 & λ + πi \end{matrix} )  (2.1)

where

R_{0}(λ) = −\frac{Γ (\frac{λ}{2πi}) Γ (\frac{1}{2} − \frac{λ}{2πi})}{Γ (−\frac{λ}{2πi}) Γ (\frac{1}{2} + \frac{λ}{2πi})},

notice that

R_{0}(λ)R_{0}(−λ) = 1

This R-matrix appears in the rational limit from XXZ R-matrix, it is related to usual XXX R-matrix by

R(λ) = (σ^{z} ⊗ I)R_{XXX}(λ)(I ⊗ σ^{z})

(similar transformation is needed when obtaining form factors of SU(2)-invariant Thirring model from SG ones [15]).

This R-matrix (2.1) satisfies the equation:

R(−πi) = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )  (2.2)

Following [5] we introduce functions g_{n} which satisfy the qKZ equations on level −4 [12]. We write the qKZ equations in their original form [15, 16] which takes into account explicitly symmetry:

g_{n}(λ_{1}, ⋯, λ_{j+1}, λ_{j}, ⋯, λ_{2n})_{ϵ_{1},···,ϵ′_{j+1},ϵ′_{j},···,ϵ_{2n}} =  (2.3)
= R(λ_{j} − λ_{j+1})_{ϵ′_{j},ϵ′_{j+1}}^{ϵ_{j},ϵ_{j+1}} g_{n}(λ_{1}, ⋯, λ_{j}, λ_{j+1}, ⋯, λ_{2n})_{ϵ_{1},···,ϵ_{j},ϵ_{j+1},···,ϵ_{2n}}
g_{n}(λ_{1}, ⋯, λ_{2n−1}, λ_{2n} + 2πi)_{ϵ_{1},···,ϵ_{2n−1},ϵ_{2n}} = g_{n}(λ_{2n}, λ_{1}, ⋯, λ_{2n−1})_{ϵ_{2n},ϵ_{1},···,ϵ_{2n−1}}  (2.4)

Solutions to these equations are meromorphic functions with possible singularities at the points

ℑ(λ_{j} − λ_{k}) = πl, l ∈ Z\0

For application to the correlation functions we are interested in a particular solution g_{n} described in details in [4, 5]. Detailed study of this solution will be performed in a future publication, in the present paper we need only limited information about it. First, the g_{n} is regular at ℑ(λ_{j} −λ_{k}) = ±π.

Moreover, much can be said about its values at these points [5] :

g_{n}(λ_{1}, ⋯, λ_{j−1}, λ_{j}, λ_{j} − πi, λ_{j+2}, ⋯, λ_{2n})_{ϵ_{1},···,ϵ_{j−1},ϵ_{j},ϵ_{j+1},ϵ_{j+2},···,ϵ_{2n}} =
= δ_{ϵ_{j},−ϵ_{j+1}} g_{n−1}(λ_{1}, ⋯, λ_{j−1}, λ_{j+2}, ⋯, λ_{2n})_{ϵ_{1},···,ϵ_{j−1},ϵ_{j+2},···,ϵ_{2n}}  (2.5)

Together with the symmetry (2.3), and (2.2) this equation implies

∑_{ϵ_{j}=−ϵ_{j+1}} g_{n}(λ_{1}, ⋯, λ_{j−1}, λ_{j}, λ_{j} + πi, λ_{j+2}, ⋯, λ_{2n})_{ϵ_{1},···,ϵ_{j−1},ϵ_{j},ϵ_{j+1},ϵ_{j+2},···,ϵ_{2n}} =

= g_{n−1}(λ_{1}, ⋯, λ_{j−1}, λ_{j+2}, ⋯, λ_{2n})_{ϵ_{1},···,ϵ_{j−1},ϵ_{j+2},···,ϵ_{2n}}  (2.6)

The emptiness formation probability P_{n} is related to g_{n} as follows

P_{n}(z_{1}, ⋯, z_{n}) = g_{n} (πz_{1}, ⋯, πz_{n}, π(z_{n} + i), ⋯, π(z_{1} + i))_{−,···,−,+,···,+}  (2.7)

Now we want to establish some general properties of P_{n} following from (2.3), (2.4), (2.5), (2.6).

1. Symmetry. The function P_{n}(z_{1}, ⋯, z_{n}) is symmetric. Proof. Obviously, it is enough to show that

P_{n}(⋯, z_{j}, z_{j+1}, ⋯) = P_{n}(⋯, z_{j+1}, z_{j}, ⋯)

This identity follows from (2.3) and from the fact that the R-matrix acts diagonally on the indices −, − and +, +:

P_{n}(⋯, z_{j}, z_{j+1}, ⋯) =
= g_{n} (⋯, πz_{j}, πz_{j+1}, ⋯, π(z_{j+1} + i), π(z_{j} + i) ⋯)_{···,−,−,···,+,+,···} =
= R_{0}(π(z_{j+1} − z_{j}))R_{0}(π(z_{j} − z_{j+1}))
× g_{n} (⋯, πz_{j+1}, πz_{j}, ⋯, π(z_{j} + i), π(z_{j+1} + i) ⋯)_{···,−,−,···,+,+,···} =
= P_{n}(⋯, z_{j+1}, z_{j}, ⋯)

QED

2. Vanishing. The function P_{n}(z_{1}, ⋯, z_{n}) vanishes when z_{k} = z_{j} + i. Proof. Due to the previous property it is sufficient to consider the case k = 1, j = 2. Let us put, first, z_{1} = z + i, z_{2} = z′, then we shall take the limit z → z′.

P_{n}(z + i, z′, ⋯, z_{n}) =
= g_{n} (π(z + i), πz′, ⋯, πz_{n}, π(z_{n} + i), ⋯, π(z′ + i), π(z + 2i))_{−,−,···,−,+,···,+,+} =
= g_{n} (πz, π(z + i), πz′, ⋯, πz_{n}, π(z_{n} + i), ⋯, π(z′ + i),)_{+,−,−,···,−,+,···,+}

where we used (2.4). Consider the limit z → z′. As it has been explained singularities do not occur for ℑ(λ_{j} − λ_{k}) = ±π. Moreover, the final result contains the fragment

g_{n}(⋯, π(z + i), πz ⋯)_{···,−,−,···}

which implies that the result vanishes due to (2.5). Because of absence of singularities this zero does not interfere with any pole, so

P_{n}(z + i, z, ⋯, z_{n}) = 0  (2.8)

QED

3. Normalization. The following asymptotic holds for z_{1} → ∞ along the real axis:

P_{n}(z_{1}, z_{2}, ⋯, z_{n}) → \frac{1}{2}P_{n−1}(z_{2}, ⋯, z_{n})  (2.9)

Proof. One more property of the solution g_{n} will be important for us. Using the integral formula from [5] one can show that g_{n}(λ_{1}, ⋯, λ_{2n−2}, λ, λ + πi) behaves as O(1) when λ → ∞ + iκ where κ is a finite number. The leading term of asymptotic does not depend on κ. We shall use notation:

g_{n}(λ_{1}, ⋯, λ_{2n−2}, λ, λ + πi)_{ϵ_{1},···,ϵ_{2n−2},ϵ_{2n−1},ϵ_{2n}} → ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; ϵ_{2n−1},ϵ_{2n}}  (2.10)

The function ĝ_{n} possesses important property of symmetry with respect to last two indices because

g_{n}(λ_{1}, ⋯, λ_{2n−2}, λ, λ + πi)_{ϵ_{1},···,ϵ_{2n−2},ϵ_{2n−1},ϵ_{2n}} =
= g_{n}(λ − πi, λ_{1}, ⋯, λ_{2n−2}, λ)_{ϵ_{2n}ϵ_{1},···,ϵ_{2n−2},ϵ_{2n−1}}
= R(λ_{1} − λ + πi)_{ϵ_{1},ϵ_{2n}}^{ϵ′_{1},σ_{2n−3}} ⋯ R(λ_{2n−2} − λ + πi)_{ϵ_{2n−2},σ_{1}}^{e′_{2n−2},ϵ′_{2n}}
× g_{n}(λ_{1}, ⋯, λ_{2n−2}, λ − πi, λ)_{ϵ′_{1},···,ϵ′_{2n−2},ϵ′_{2n},ϵ_{2n−1}} →
→ sign(ϵ_{2n−1}, ϵ_{2n}) ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; ϵ_{2n},ϵ_{2n−1}}  (2.11)

with the sign function

sign(ϵ_{1}, ϵ_{2}) = \{ \begin{matrix}−1 & if ϵ_{1} = ϵ_{2} \\ 1 & if ϵ_{1} = −ϵ_{2} \end{matrix}  (2.12)

where we used the asymptotic of the R-matrix

R(λ) ⟶_{λ→∞} (−i) · diag(1, −1, −1, 1)

From the equation (2.11) we conclude that

ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; ϵ_{2n−1},ϵ_{2n}} = 0 if ϵ_{2n−1} = ϵ_{2n}
ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; +,−} = ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; −,+}  (2.13)

The relation (2.13) allows to calculate ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; +,−}:

g_{n−1}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2}} =
= ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; +,−} + ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; −,+} =
= 2 ĝ_{n}(λ_{1}, ⋯, λ_{2n−2})_{ϵ_{1},···,ϵ_{2n−2} ; +,−}

Now the normalization (2.9) follows from

P_{n}(z_{1}, z_{2}, ⋯, z_{n}) =
= g_{n} (πz_{1}, πz_{2}, ⋯, πz_{n}, π(z_{n} + i), ⋯, π(z_{2} + i), π(z_{1} + i))_{−,−,···,−,+,···,+,+} =
= g_{n} (πz_{2}, ⋯, πz_{n}, π(z_{n} + i), ⋯, π(z_{2} + i), π(z_{1} + i), π(z_{1} + 2i))_{−,···,−,+,···,+,+,−}
⟶_{z_{1}→∞} ĝ_{n} (πz_{2}, ⋯, πz_{n}, π(z_{n} + i), ⋯, π(z_{2} + i))_{−,···,−,+,···,+ ; +,−} =
= \frac{1}{2} P_{n−1}(z_{2}, ⋯, z_{n})  (2.14)

where the formula (2.10) was used. QED The above three properties will be very useful for further consideration.

3. Explicit expressions for EFP.

In paper [6] Jimbo and Miwa suggested an integral representation as a solution to the qKZ (2.3) - (2.4) which also satisfies the property (2.5). Using the relation (2.7) one gets the integral representation which is the direct generalization of the formula (1.3) to the inhomogeneous case

P_{n}(z_{1}, …, z_{n}) = π^{\frac{n(n+1)}{2}} ∏_{k<j} \frac{sinh (π(z_{k} − z_{j}))}{π(z_{k} − z_{j})} ∫_{−i/2−∞}^{−i/2+∞} \frac{dλ_{1}}{2πi} … ∫_{−i/2−∞}^{−i/2+∞} \frac{dλ_{n}}{2πi} ∏_{k<j} \frac{sinh(π(λ_{j} − λ_{k}))}{(λ_{j} − λ_{k} − i)}·

· \frac{∏_{j=1}^{n}(∏_{k=1}^{j−1}(λ_{j} − z_{k} − i) ∏_{k=j+1}^{n}(λ_{j} − z_{k}))}{∏_{j=1}^{n} ∏_{k=1}^{n} sinh(π(λ_{j} − z_{k}))}  (3.1)

where for the moment we will consider the inhomogeneity parameters z_{j} as distinct real numbers. These parameters also may be considered as arbitrary complex numbers, but the integration contours in this case should be taken in such a way that they separate the singularities of the integrand in the same manner as the contours in the above formula (3.1) for the real values z_{j}.

For evaluation of the integral (3.1) we can use the technique described in [9]. We obtain for the first four values of P_{n}(z_{1}, …, z_{n}) :

P_{1} = \frac{1}{2}  (3.2)
P_{2} = \frac{1}{3} + A_{2,1}(z_{1}, z_{2}) G(z_{1} − z_{2})  (3.3)
P_{3} = \frac{1}{4} + A_{3,1}(z_{1}, z_{2}|z_{3})G(z_{1} − z_{2}) + A_{3,1}(z_{1}, z_{3}|z_{2})G(z_{1} − z_{3}) + A_{3,1}(z_{2}, z_{3}|z_{1})G(z_{2} − z_{3}) (3.4)

P_{4} = \frac{1}{5} + A_{4,1}(z_{1}, z_{2}|z_{3}, z_{4})G(z_{1} − z_{2}) + A_{4,1}(z_{1}, z_{3}|z_{2}, z_{4}) G(z_{1} − z_{3}) + A_{4,1}(z_{1}, z_{4}|z_{2}, z_{3}) G(z_{1} − z_{4})+

A_{4,1}(z_{2}, z_{3}|z_{1}, z_{4}) G(z_{2} − z_{3}) + A_{4,1}(z_{2}, z_{4}|z_{1}, z_{3}) G(z_{2} − z_{4}) + A_{4,1}(z_{3}, z_{4}, |z_{1}, z_{2}) G(z_{3} − z_{4})+
A_{4,2}(z_{1}, z_{2}|z_{3}, z_{4})G(z_{1} − z_{2})G(z_{3} − z_{4}) + A_{4,2}(z_{1}, z_{3}|z_{2}, z_{4})G(z_{1} − z_{3})G(z_{2} − z_{4})+
A_{4,2}(z_{1}, z_{4}|z_{2}, z_{3})G(z_{1} − z_{4})G(z_{2} − z_{3})  (3.5)

where

G(x) = (x^{2} + 1)(β(ix) + β(−ix)) = 2 ∑_{k=1}^{∞}(−1)^{k} · k · \frac{x^{2} + 1}{x^{2} + k^{2}}  (3.6)

β(x) = \frac{1}{2}(ψ(\frac{1 + x}{2}) − ψ(\frac{x}{2})) = ∑_{k=0}^{∞} \frac{(−1)^{k}}{x + k}

ψ(x) = \frac{d}{dx} ln Γ(x)

and the functions

A_{2,1}(z_{1}, z_{2}) = \overline{Q}_{2,1}(z_{1}, z_{2})

A_{3,1}(z_{1}, z_{2}|z_{3}) = \frac{\overline{Q}_{3,1}(z_{1}, z_{2}|z_{3})}{z_{13}z_{23}}

A_{4,2}(z_{1}, z_{2}|z_{3}, z_{4}) = \frac{\overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{4})}{z_{13}z_{14}z_{23}z_{24}}

A_{4,1}(z_{1}, z_{2}|z_{3}, z_{4}) = \frac{\overline{Q}_{4,1}(z_{1}, z_{2}|z_{3}, z_{4})}{z_{13}z_{14}z_{23}z_{24}}

(3.7)

with the polynomials

\overline{Q}_{2,1}(z_{1}, z_{2}) = \frac{1}{6}  (3.8)
\overline{Q}_{3,1}(z_{1}, z_{2}|z_{3}) = \frac{1}{12}( z_{13}z_{23} − 1)  (3.9)
\overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{4}) = \frac{1}{36}\{(z_{13}z_{24} − 1)(z_{14}z_{23} − 1) + \frac{2}{5}(z_{12}^{2} + \frac{3}{2})(z_{34}^{2} + \frac{3}{2}) + \frac{3}{2}\}  (3.10)
\overline{Q}_{4,1}(z_{1}, z_{2}|z_{3}, z_{4}) = 2 \overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{4}) − \frac{1}{60}(z_{12}^{2} + 4)(z_{34}^{2} + 1)  (3.11)

and by definition

z_{jk} = z_{j} − z_{k}

The function G(x) (3.6) is actually real for real x and even

G(−x) = G(x)  (3.12)

Besides, for small x one has an expansion

G(x) = −2(1 + x^{2}) ∑_{k=0}^{∞}(−1)^{k}x^{2k}ζ_{a}(2k + 1)  (3.13)

with the alternating zeta series defined in (4.1). In particular,

G(0) = −2 ln 2  (3.14)

The following limit is well defined

G(±∞) = −\frac{1}{2}  (3.15)

We shall also need the following relation

G(x − i) = α(x) + γ(x) G(x)  (3.16)

where

α(x) = −\frac{(x − 2i)}{(x − i)}, γ(x) = −\frac{x(x − 2i)}{(x + i)(x − i)}  (3.17)

In particular,

G(± i) = −2  (3.18)

Looking at the answers (3.2-3.5) we can conclude that they satisfy the general three properties from the previous section and one more property, namely, the translational invariance:

P_{n}(z_{1} + a, …, z_{n} + a) = P_{n}(z_{1}, …, z_{n})  (3.19)

which actually follows from the integral representation (3.1). This property means that P_{n} depends only on differences of n parameters z_{j}.

One can also suggest a general ansatz :

P_{n}(z_{1}, …, z_{n}) = ∑_{l=0}^{[ \frac{n}{2} ]}\{A_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n}) ∏_{j=1}^{l} G(z_{2j−1} − z_{2j}) + permutations\}

(3.20)

where A_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n}) are some rational functions which depend only on differences z_{j}−z_{k}. We also imply that these functions are symmetric under independent transpositions z_{1} ↔ z_{2}, z_{3} ↔ z_{4}, …, z_{2l−1} ↔ z_{2l} and any permutation of the residual variables z_{2l+1} … z_{n}. Also we require a symmetry of this function under transposition of any pair z_{2r−1}z_{2r} ↔ z_{2s−1}z_{2s} for r, s ≤ l. It is implied that the permutations in (3.20) do not involve those sets of variables for which the function A_{n,l} is already symmetric. We also take

A_{n,0}(z_{1}, …, z_{n}) = \frac{1}{n + 1}  (3.21)

and ^{3}

A_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n}) = \frac{Q_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n})}{∏_{1≤j<k≤n}(z_{j} − z_{k})}  (3.22)

where Q_{n,l} are some polynomials of maximum power n − 1 for each variable z_{k}. Moreover we expect that these polynomials have factorized form, namely,

Q_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n}) = (z_{1} − z_{2})(z_{3} − z_{4}) … (z_{2l−1} − z_{2l}) ∏_{2l+1≤j<k≤n}(z_{j} − z_{k})·

\overline{Q}_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n})  (3.23)

where the polynomials \overline{Q}_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n}) depend only on differences of variables z_{j} − z_{k}. Also they have the same symmetry properties as the function A_{n,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{n}) described above. The maximum powers for each variable may be readily calculated from the corresponding power of the polynomial Q_{nl}.

Let us rewrite the properties 2 and 3 from the Section 2 in the following form

P_{n}(z_{1}, …, z_{n−2}, z_{n−1}, z_{n−1} ± i) = 0  (3.24)
lim_{z_{n}→∞} P_{n}(z_{1}, …, z_{n−1}, z_{n}) = \frac{1}{2}P_{n−1}(z_{1}, …, z_{n−1})  (3.25)

As we shall see below together with our main ansatz (3.20) they completely fix the answer for P_{5}(z_{1}, …, z_{5}) and P_{6}(z_{1}, …, z_{6}). Therefore we may expect that it also happens for any P_{n}(z_{1}, …, z_{n}).

Let us start our consideration from the analysis of the relation (3.25). First let us make a simple observation that it has an obvious solution corresponding to the case when all the variables z_{j} go to infinity ^{4} i.e.

P_{n}(∞, …, ∞) = \frac{1}{2^{n}}  (3.26)

Now we can analyze corollaries of the relation (3.25) and our general ansatz (3.20). It can be seen that this is equivalent to the following chain of recurrent relations for the functions A_{n,l}

A_{2k+1,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{2k}, ∞)−

\frac{1}{2}\{A_{2k+1,l+1}(z_{1}, z_{2}|z_{3}, z_{4}| … |∞, z_{2l+1}|z_{2l+2}, …, z_{2k}) + permutations\} =

\frac{1}{2}A_{2k,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l+1}, …, z_{2k}) for l = 0, 1, …, k  (3.27)
A_{2k,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l−1}, z_{2l}|z_{2l+1} … z_{2k−1}, ∞)−

\frac{1}{2}\{A_{2k,l+1}(z_{1}, z_{2}|z_{3}, z_{4}| … |∞, z_{2l+1}|z_{2l+2}, …, z_{2k−1}) + permutations\} =

\frac{1}{2}A_{2k−1,l}(z_{1}, z_{2}|z_{3}, z_{4}| … |z_{2l+1}, …, z_{2k−1}) for l = 0, 1, …, k − 1  (3.28)

3Since A_{n,0} are just constants below we will not write their arguments any more.

4The limits z_{n} → ∞, …, z_{1} → ∞ should be taken consequently

where permutations are taken only for the variables z_{2l+1}, z_{2l+2}, … z_{2k} for (3.27) and z_{2l+1}, z_{2l+2}, … z_{2k−1} for (3.28) in such a way that the variables within symmetric sets are not permuted. In the formula (3.27) it is also implied that

A_{2k+1,k+1} = 0.  (3.29)

Up to the moment it is not completely clear how to write down in a closed form all the equations for the functions A_{n,k} which follow from the relation (3.24).

Let us write them down together with the equations (3.27) and (3.28) consequently for n = 2, 3, 4, 5, 6.

The case n = 2 Since k = 1 in this case we have only one equation in the set (3.28)

A_{2,0} − \frac{1}{2}A_{2,1}(∞, z_{1}) = \frac{1}{2}A_{1,0}  (3.30)

which is obvious because as it can be seen from eqs. (3.2-3.3), (3.7) and (3.8) all the functions A here are just numbers, namely,

A_{1,0} = \frac{1}{2}

A_{2,0} = \frac{1}{3}, A_{2,1}(z_{1}, z_{2}) = \frac{1}{6}

The second property, namely, the formula (3.24) is also trivially satisfied because using eq. (3.17), (3.18) we have from (3.3)

P_{2}(z_{1}, z_{1} + i) = A_{2,0} + A_{2,1}(z_{1}, z_{1} + i)G(−i) =
A_{2,0} − 2 A_{2,1}(z_{1}, z_{1} + i) = 0  (3.31)

So the equation corresponding to the second property is just the last relation in formula (3.31).

The case n = 3 Since n is odd the first property (3.25) corresponds to the relations (3.27) with k = 1. Here we have two of them

A_{3,1}(z_{1}, z_{2}|∞) = \frac{1}{2}A_{2,1}(z_{1}, z_{2})

A_{3,0}(z_{1}, z_{2}|∞) − \frac{1}{2}\{A_{3,1}(∞, z_{1}|z_{2}) + A_{3,1}(∞, z_{2}|z_{1})\} = \frac{1}{2}A_{2,1}(z_{1}, z_{2})  (3.32)

The second property (3.24) i.e. the equation

P_{3}(z_{1}, z_{2}, z_{2} + i) = 0  (3.33)

is equivalent to the following two relations

A_{3,1}(z_{1}, z_{2}|z_{2} + i) + γ(z_{12})A_{3,1}(z_{1}, z_{2} + i|z_{2}) = 0
A_{3,0} + α(z_{12})A_{3,1}(z_{1}, z_{2} + i|z_{2}) − 2 A_{3,1}(z_{2}, z_{2} + i|z_{1}) = 0  (3.34)

Substituting here A_{3,1} defined in (3.7) and (3.9) (A_{3,0} = 1/4) one can easily check that these two equations are satisfied.

The case n = 4 As in the previous case the first property (3.25) is equivalent to the set of two relations (see (3.28) for k = 2)

A_{4,1}(z_{1}, z_{2}|z_{3}, ∞) − \frac{1}{2}A_{4,2}(z_{1}, z_{2}|∞, z_{3}) = \frac{1}{2}A_{3,1}(z_{1}, z_{2}|z_{3})

A_{4,0} − \frac{1}{2}\{A_{4,1}(z_{1}, ∞|z_{2}, z_{3}) + A_{4,1}(z_{2}, ∞|z_{1}, z_{3}) + A_{4,1}(z_{3}, ∞|z_{1}, z_{2})\} = \frac{1}{2}A_{3,0}  (3.35)

The Property 2 (3.24) for n = 4 is equivalent to the following set of equations

γ(z_{23})A_{4,2}(z_{1}, z_{3}|z_{2}, z_{3} + i) + γ(z_{13})A_{4,2}(z_{1}, z_{3} + i|z_{2}, z_{3}) = 0
A_{4,1}(z_{1}, z_{2}|z_{3}, z_{3} + i) − 2 A_{4,2}(z_{1}, z_{2}|z_{3}, z_{3} + i) = 0
A_{4,1}(z_{1}, z_{3}|z_{2}, z_{3} + i) + γ(z_{13})A_{4,1}(z_{1}, z_{3} + i|z_{2}, z_{3}) + α(z_{23})A_{4,2}(z_{1}, z_{3}|z_{2}, z_{3} + i) = 0
A_{4,0} + α(z_{13})A_{4,1}(z_{1}, z_{3} + i|z_{2}, z_{3}) + α(z_{23})A_{4,1}(z_{2}, z_{3} + i|z_{1}, z_{3}) − 2 A_{4,1}(z_{3}, z_{3} + i|z_{1}, z_{2}) = 0 (3.36)

Again using eqs. (3.7) and (3.10-3.11) one can check that the equations (3.35-3.36) are satisfied. So we have established that, in fact, the formulae (3.7) with the definitions (3.8-3.11) provide the

solution to the equations (3.30) and (3.31) for n = 2, eqs. (3.32) and (3.34) for n = 3, eqs. (3.35) and (3.36) for n = 4.

We may also go an opposite way, namely, demand all these equations, solve them in framework of the ansatz (3.20-3.23) and then come to the result for P_{n}. Let us mention that this way seems to be much easier than the direct calculation of the integral (3.1) as we did before.

For n = 5, 6 we would like to try this new way of getting the result for EFP.

The case n = 5 First as above we write down consequences of the properties (3.25) and (3.24). The first one (3.25) is equivalent to the set of three relations which can be obtained from eqs. (3.27) by taking k = 2

A_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|∞) = \frac{1}{2} A_{4,2}(z_{1}, z_{2}|z_{3}, z_{4})

A_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, ∞) − \frac{1}{2}\{A_{5,2}(z_{1}, z_{2}|∞, z_{3}|z_{4}) + A_{5,2}(z_{1}, z_{2}|∞, z_{4}|z_{3})\} = \frac{1}{2}A_{4,1}(z_{1}, z_{2}|z_{3}, z_{4})

A_{5,0} − \frac{1}{2}\{A_{5,1}(∞, z_{1}|z_{2}, z_{3}, z_{4}) + A_{5,1}(∞, z_{2}|z_{1}, z_{3}, z_{4}) +

+ A_{5,1}(∞, z_{3}|z_{1}, z_{2}, z_{4}) + A_{5,1}(∞, z_{4}|z_{1}, z_{2}, z_{3})\} = \frac{1}{2}A_{4,0}  (3.37)

The Property 2 (3.24) for n = 5 is equivalent to the following set of five equations

A_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{4} + i) + γ(z_{34})A_{5,2}(z_{1}, z_{2}|z_{3}, z_{4} + i|z_{4}) = 0
γ(z_{14})A_{5,2}(z_{1}, z_{4} + i|z_{2}, z_{4}|z_{3}) + γ(z_{24})A_{5,2}(z_{1}, z_{4}|z_{2}, z_{4} + i|z_{3}) = 0

α(z_{34})A_{5,2}(z_{1}, z_{2}|z_{3}, z_{4} + i|z_{4}) − 2 A_{5,2}(z_{1}, z_{2}|z_{4} + i, z_{4}|z_{3}) + A_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{4} + i) = 0

α(z_{24})A_{5,2}(z_{1}, z_{4}|z_{2}, z_{4} + i|z_{3}) + α(z_{34})A_{5,2}(z_{1}, z_{4}|z_{4} + i, z_{3}|z_{2}) + A_{5,1}(z_{1}, z_{4}|z_{2}, z_{3}, z_{4} + i) +

γ(z_{14})A_{5,1}(z_{1}, z_{4} + i|z_{2}, z_{3}, z_{4}) = 0
A_{5,0} + α(z_{14})A_{5,1}(z_{1}, z_{4} + i|z_{2}, z_{3}, z_{4}) + α(z_{24})A_{5,1}(z_{2}, z_{4} + i|z_{1}, z_{3}, z_{4}) +
+ α(z_{34})A_{5,1}(z_{3}, z_{4} + i|z_{1}, z_{2}, z_{4}) − 2 A_{5,1}(z_{4}, z_{4} + i|z_{1}, z_{2}, z_{3}) = 0  (3.38)

The difference with above cases n = 2, 3, 4 is that now up to the moment we do not have a solution to these equations. Therefore we have to get it. Let us briefly describe how we do this. First we take the ansatz (3.21-3.23), namely,

A_{5,0} = \frac{1}{6}  (3.39)
A_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}) = \frac{\overline{Q}_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5})}{z_{13}z_{14}z_{15}z_{23}z_{24}z_{25}}  (3.40)
A_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}) = \frac{\overline{Q}_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5})}{z_{13}z_{14}z_{15}z_{23}z_{24}z_{25}z_{35}z_{45}}  (3.41)

and demand that the polynomials \overline{Q}_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}) and \overline{Q}_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}) have the same symmetry properties as the functions A_{5,1} and A_{5,2} and the same maximum powers of the variables z_{1}, …, z_{5} as their denominators in the r.h.s. of (3.40) and (3.41) respectively. Namely, we write

\overline{Q}_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}) = ∑ ; \begin{matrix}0 ≤ i_{1}, i_{2} ≤ 3, 0 ≤ i_{3}, i_{4}, i_{5} ≤ 2 \\ i_{1} + … + i_{5} ≤ 6 \end{matrix} C_{5,1}(i_{1}, i_{2}|i_{3}, i_{4}, i_{5}) z_{1}^{i_{1}}z_{2}^{i_{2}}z_{3}^{i_{3}}z_{4}^{i_{4}}z_{5}^{i_{5}} (3.42)

\overline{Q}_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}) = ∑ ; \begin{matrix}0 ≤ i_{1}, i_{2}, i_{3}, i_{4} ≤ 3, 0 ≤ i_{5} ≤ 4 \\ i_{1} + … + i_{5} ≤ 8 \end{matrix} C_{5,2}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}) z_{1}^{i_{1}}z_{2}^{i_{2}}z_{3}^{i_{3}}z_{4}^{i_{4}}z_{5}^{i_{5}} (3.43)

with the coefficients C_{5,1}(i_{1}, i_{2}|i_{3}, i_{4}, i_{5}) and C_{5,2}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}) which have evident symmetry properties

C_{5,1}(i_{1}, i_{2}|i_{3}, i_{4}, i_{5}) = C_{5,1}(i_{2}, i_{1}|i_{3}, i_{4}, i_{5}) = C_{5,1}(i_{1}, i_{2}|σ(i_{3}), σ(i_{4}), σ(i_{5}))  (3.44)

where σ is any element of the permutation group of three elements S_{3} while

C_{5,2}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}) = C_{5,2}(i_{2}, i_{1}|i_{3}, i_{4}|i_{5}) = C_{5,2}(i_{1}, i_{2}|i_{4}, i_{3}|i_{5}) = C_{5,2}(i_{3}, i_{4}|i_{1}, i_{2}|i_{5})  (3.45)

In accordance with eq. (3.19) we also demand the translational invariance of these polynomials

\overline{Q}_{5,1}(z_{1} + a, z_{2} + a|z_{3} + a, z_{4} + a, z_{5} + a) = \overline{Q}_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5})  (3.46)
\overline{Q}_{5,2}(z_{1} + a, z_{2} + a|z_{3} + a, z_{4} + a|z_{5} + a) = \overline{Q}_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5})  (3.47)

These conditions allow us to fix some of the coefficients C_{5,1} and C_{5,2} in (3.42-3.43).

The rest of these coefficients may be fixed by satisfying the equations (3.38). One should say that in spite of the fact that it turns out to be an over determined system it has a solution and this solution is unique. Also it is interesting to note that after the equations (3.38) are satisfied all the residual equations (3.37) are satisfied automatically (!).

Let us show the result

\overline{Q}_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}) =
\frac{1}{2}(z_{15}z_{25} − 1)(z_{35}z_{45} − 1) \overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{4}) + \frac{1}{360}(z_{12}^{2} + 4)(z_{34}^{2} + 4)(5 − z_{13}z_{24} − z_{14}z_{23}) (3.48)
\overline{Q}_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}) =

\frac{1}{2}(z_{15}z_{25} − 1) \overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{4}) + \frac{1}{2}(z_{14}z_{24} − 1) \overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{5}) + \frac{1}{2}(z_{13}z_{23} − 1) \overline{Q}_{4,2}(z_{1}, z_{2}|z_{4}, z_{5})

− \frac{(z_{12}^{2} + 4)}{360}\{22 + (z_{34}^{2} − 2)(z_{15}z_{25} + \frac{1}{2}) + (z_{35}^{2} − 2)(z_{14}z_{24} + \frac{1}{2}) + (z_{45}^{2} − 2)(z_{13}z_{23} + \frac{1}{2})\} (3.49)

Now using eqs. (3.39-3.41) and substituting these formulae into the ansatz (3.20) we get the answer for P_{5} in the inhomogeneous case.

The case n = 6 We proceed in the same line as for the case n = 5. The Property 1 i.e. eq.(3.25) or equivalently the set of the recurrent relations (3.28) at k = 3 gives us three equations

A_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, ∞) − \frac{1}{2}A_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, ∞) = \frac{1}{2}A_{5,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5})

A_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, ∞) − \frac{1}{2}\{A_{6,2}(z_{1}, z_{2}|∞, z_{3}|z_{4}, z_{5}) + A_{6,2}(z_{1}, z_{2}|∞, z_{4}|z_{3}, z_{5}) +

+ A_{6,2}(z_{1}, z_{2}|∞, z_{5}|z_{3}, z_{4})\} = \frac{1}{2}A_{5,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5})

A_{6,0} − \frac{1}{2}\{A_{6,1}(∞, z_{1}|z_{2}, z_{3}, z_{4}, z_{5}) + A_{6,1}(∞, z_{2}|z_{1}, z_{3}, z_{4}, z_{5}) + A_{6,1}(∞, z_{3}|z_{1}, z_{2}, z_{4}, z_{5}) +

+ A_{6,1}(∞, z_{4}|z_{1}, z_{2}, z_{3}, z_{5}) + A_{6,1}(∞, z_{5}|z_{1}, z_{2}, z_{3}, z_{4})\} = \frac{1}{2}A_{50}  (3.50)

The Property 2 (3.24) for n = 6 produces seven equations

A_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{5} + i) − 2 A_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{5} + i) = 0

γ(z_{45})A_{6,3}(z_{1}, z_{2}|z_{3}, z_{5}|z_{4}, z_{5} + i) + γ(z_{35})A_{6,3}(z_{1}, z_{2}|z_{3}, z_{5} + i|z_{4}, z_{5}) = 0
A_{6,2}(z_{1}, z_{2}|z_{3}, z_{5}|z_{4}, z_{5} + i) + γ(z_{35})A_{6,2}(z_{1}, z_{2}|z_{3}, z_{5} + i|z_{4}, z_{5}) +
+ α(z_{45})A_{6,3}(z_{1}, z_{2}|z_{3}, z_{5}|z_{4}, z_{5} + i) = 0
α(z_{35})A_{6,2}(z_{1}, z_{2}|z_{3}, z_{5} + i|z_{4}, z_{5}) + α(z_{45})A_{6,2}(z_{1}, z_{2}|z_{4}, z_{5} + i|z_{3}, z_{5}) −

− 2 A_{6,2}(z_{1}, z_{2}|z_{5}, z_{5} + i|z_{3}, z_{4}) + A_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{5} + i) = 0
γ(z_{25})A_{6,2}(z_{1}, z_{5}|z_{2}, z_{5} + i|z_{3}, z_{4}) + γ(z_{15})A_{6,2}(z_{1}, z_{5} + i|z_{2}, z_{5}|z_{3}, z_{4}) = 0
A_{6,1}(z_{1}, z_{5}|z_{2}, z_{3}, z_{4}, z_{5} + i) + γ(z_{15})A_{6,1}(z_{1}, z_{5} + i|z_{2}, z_{3}, z_{4}, z_{5}) +
α(z_{25})A_{6,2}(z_{1}, z_{5}|z_{2}, z_{5} + i|z_{3}, z_{4}) + α(z_{35})A_{6,2}(z_{1}, z_{5}|z_{3}, z_{5} + i|z_{2}, z_{4}) +
+ α(z_{45})A_{6,2}(z_{1}, z_{5}|z_{4}, z_{5} + i|z_{2}, z_{3}) = 0
A_{6,0} − 2 A_{6,1}(z_{5}, z_{5} + i|z_{1}, z_{2}, z_{3}, z_{4}) + α(z_{15})A_{6,1}(z_{1}, z_{5} + i|z_{2}, z_{3}, z_{4}, z_{5}) +
α(z_{25})A_{6,1}(z_{2}, z_{5} + i|z_{1}, z_{3}, z_{4}, z_{5}) + α(z_{35})A_{6,1}(z_{3}, z_{5} + i|z_{1}, z_{2}, z_{4}, z_{5}) +
+ α(z_{45})A_{6,1}(z_{4}, z_{5} + i|z_{1}, z_{2}, z_{3}, z_{4}) = 0  (3.51)

In the same line as we did it above for the case n = 5 we can solve the equations (3.50) and (3.51) in framework of the ansatz (3.21-3.23)

A_{6,0} = \frac{1}{7}  (3.52)
A_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6}) = \frac{\overline{Q}_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6})}{z_{13}z_{14}z_{15}z_{16}z_{23}z_{24}z_{25}z_{26}}  (3.53)

A_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) = \frac{\overline{Q}_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6})}{z_{13}z_{14}z_{15}z_{16}z_{23}z_{24}z_{25}z_{26}z_{35}z_{36}z_{45}z_{46}}  (3.54)

A_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) = \frac{\overline{Q}_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6})}{z_{13}z_{14}z_{15}z_{16}z_{23}z_{24}z_{25}z_{26}z_{35}z_{36}z_{45}z_{46}}  (3.55)

As above we demand that the polynomials \overline{Q}_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6}), \overline{Q}_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) and \overline{Q}_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) have the same symmetry properties as the functions A_{6,1}, A_{6,2} , A_{6,3} respectively. Analogous to eqs. (3.42-3.43) we write

\overline{Q}_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6}) = ∑ ; \begin{matrix}0 ≤ i_{1}, i_{2} ≤ 4, 0 ≤ i_{3}, i_{4}, i_{5}, i_{6} ≤ 2 \\ i_{1} + … + i_{6} ≤ 8 \end{matrix} C_{6,1}(i_{1}, i_{2}|i_{3}, i_{4}, i_{5}, i_{6}) z_{1}^{i_{1}}z_{2}^{i_{2}}z_{3}^{i_{3}}z_{4}^{i_{4}}z_{5}^{i_{5}}z_{6}^{i_{6}}

(3.56)

\overline{Q}_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) = ∑ ; \begin{matrix}0 ≤ i_{1}, i_{2}, …, i_{6} ≤ 4 \\ i_{1} + … + i_{6} ≤ 12 \end{matrix} C_{6,2}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}, i_{6}) z_{1}^{i_{1}}z_{2}^{i_{2}}z_{3}^{i_{3}}z_{4}^{i_{4}}z_{5}^{i_{5}}z_{6}^{i_{6}}  (3.57)

\overline{Q}_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) = ∑ ; \begin{matrix}0 ≤ i_{1}, i_{2}, …, i_{6} ≤ 4 \\ i_{1} + … + i_{6} ≤ 12 \end{matrix} C_{6,3}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}, i_{6}) z_{1}^{i_{1}}z_{2}^{i_{2}}z_{3}^{i_{3}}z_{4}^{i_{4}}z_{5}^{i_{5}}z_{6}^{i_{6}}  (3.58)

with the coefficients C_{6,1}(i_{1}, i_{2}|i_{3}, i_{4}, i_{5}, i_{6}), C_{6,2}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}, i_{6}) and C_{6,3}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}, i_{6}) which satisfy the symmetry conditions

C_{6,1}(i_{1}, i_{2}|i_{3}, i_{4}, i_{5}, i_{6}) = C_{6,1}(i_{2}, i_{1}|i_{3}, i_{4}, i_{5}, i_{6}) = C_{6,1}(i_{1}, i_{2}|σ(i_{3}), σ(i_{4}), σ(i_{5}), σ(i_{6}))  (3.59)

where σ is any element of the permutation group of four elements S_{4}

C_{6,2}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}, i_{6}) = C_{6,2}(i_{2}, i_{1}|i_{3}, i_{4}|i_{5}, i_{6}) = C_{6,2}(i_{1}, i_{2}|i_{4}, i_{3}|i_{5}, i_{6}) = C_{6,2}(i_{1}, i_{2}|i_{3}, i_{4}|i_{6}, i_{5}) =

= C_{6,2}(i_{3}, i_{4}|i_{1}, i_{2}|i_{5}, i_{6})  (3.60)

C_{6,3}(i_{1}, i_{2}|i_{3}, i_{4}|i_{5}, i_{6}) = C_{6,3}(i_{2}, i_{1}|i_{3}, i_{4}|i_{5}, i_{6}) = C_{6,3}(i_{1}, i_{2}|i_{4}, i_{3}|i_{5}, i_{6}) = C_{6,3}(i_{1}, i_{2}|i_{3}, i_{4}|i_{6}, i_{5}) =

= C_{6,3}(i_{3}, i_{4}|i_{1}, i_{2}|i_{5}, i_{6}) = C_{6,3}(i_{1}, i_{2}|i_{5}, i_{6}|i_{3}, i_{4})  (3.61)

As in the previous case the translational invariance of these polynomials

\overline{Q}_{6,1}(z_{1} + a, z_{2} + a|z_{3} + a, z_{4} + a, z_{5} + a, z_{6} + a) = \overline{Q}_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6})  (3.62)
\overline{Q}_{6,2}(z_{1} + a, z_{2} + a|z_{3} + a, z_{4} + a|z_{5} + a, z_{6} + a) = \overline{Q}_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6})  (3.63)
\overline{Q}_{6,3}(z_{1} + a, z_{2} + a|z_{3} + a, z_{4} + a|z_{5} + a, z_{6} + a) = \overline{Q}_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6})  (3.64)

allow us to fix a lot of the coefficients C_{6,1}, C_{6,2} and C_{6,3}.

Then with the help of computer we solve the over determined system of the equations (3.51). It is left to check that all other equations (3.50) are satisfied automatically as in the case n = 5.

The result looks as follows

\overline{Q}_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) =

6^{3} \overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{4})\overline{Q}_{4,2}(z_{1}, z_{2}|z_{5}, z_{6})\overline{Q}_{4,2}(z_{3}, z_{4}|z_{5}, z_{6}) + (z_{12}^{2}+4)(z_{34}^{2}+4)(z_{56}^{2}+4) Λ_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6})

(3.65)

\overline{Q}_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) =

\frac{6^{3}}{2} \overline{Q}_{4,2}(z_{1}, z_{2}|z_{3}, z_{4})\overline{Q}_{4,1}(z_{1}, z_{2}|z_{5}, z_{6})\overline{Q}_{4,1}(z_{3}, z_{4}|z_{5}, z_{6}) + (z_{12}^{2} + 4)(z_{34}^{2} + 4) Λ_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6})

(3.66)

\overline{Q}_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6}) =

\frac{1}{30} (z_{13}z_{23} − 1)(z_{14}z_{24} − 1)(z_{15}z_{25} − 1)(z_{16}z_{26} − 1) + (z_{12}^{2} + 4) Λ_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6})  (3.67)

where

Λ_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) = \frac{1}{5400}(5 − z_{13}z_{24} − z_{14}z_{23})(5 − z_{15}z_{26} − z_{16}z_{25})(5 − z_{35}z_{46} − z_{36}z_{45})

− \frac{1}{189000}(z_{12}^{2} − 26)(z_{34}^{2} − 26)(z_{56}^{2} − 26) + \frac{1}{200}(z_{12}^{2} + z_{34}^{2} + z_{56}^{2} − 16)  (3.68)

Λ_{6,2}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) = \{\frac{1}{2}Λ_{6,3}(z_{1}, z_{2}|z_{3}, z_{4}|z_{5}, z_{6}) + \frac{1}{100}\overline{Q}_{42}(z_{1}, z_{2}|z_{3}, z_{4})(z_{56}^{2} + 1)\}(z_{56}^{2} + 13) +

\{\frac{1}{720}(5 − z_{13}z_{24} − z_{14}z_{23})(z_{56}^{2} − 3 + z_{15}z_{26} + z_{16}z_{25} + z_{35}z_{46} + z_{36}z_{45}) −

− \frac{1}{2520}(z_{12}^{2} + \frac{11}{2})(z_{34}^{2} + \frac{11}{2}) − \frac{1}{160}\}(z_{56}^{2} + 1)  (3.69)

Λ_{6,1}(z_{1}, z_{2}|z_{3}, z_{4}, z_{5}, z_{6}) = \frac{1}{600}(z_{13}z_{14}z_{23}z_{24}z_{56}^{2} + z_{13}z_{15}z_{23}z_{25}z_{46}^{2} + … + z_{15}z_{16}z_{25}z_{26}z_{34}^{2}) +

+ \frac{11}{900}(z_{13}z_{14}z_{23}z_{24} + z_{13}z_{15}z_{23}z_{25} + … + z_{15}z_{16}z_{25}z_{26}) −

− \frac{1}{225}\{(z_{13}z_{24} + z_{14}z_{23})z_{56}^{2} + (z_{13}z_{25} + z_{15}z_{23})z_{46}^{2} + … + (z_{15}z_{26} + z_{16}z_{25})z_{34}^{2}\} −

− \frac{7}{300}\{(z_{13}z_{24}+z_{14}z_{23}) + (z_{13}z_{25}+z_{15}z_{23}) + … + (z_{15}z_{26}+z_{16}z_{25})\} + \frac{1}{8400}z_{12}^{2}(z_{34}^{2}z_{56}^{2}+z_{35}^{2}z_{46}^{2}+z_{36}^{2}z_{45}^{2}) −

− \frac{19}{2520}(z_{34}^{2}z_{56}^{2} + z_{35}^{2}z_{46}^{2} + z_{36}^{2}z_{45}^{2}) + \frac{1}{1575}z_{12}^{2}(z_{34}^{2} + z_{56}^{2} + z_{35}^{2} + z_{46}^{2} + z_{36}^{2} + z_{45}^{2}) −

− \frac{1}{168}(z_{34}^{2} + z_{56}^{2} + z_{35}^{2} + z_{46}^{2} + z_{36}^{2} + z_{45}^{2}) + \frac{13}{840}z_{12}^{2} + \frac{29}{70}  (3.70)

By means of this solution, eqs. (3.52-3.55) and the ansatz (3.20) for n = 6 we come (for the very first time (!)) to the result for P_{6} in the inhomogeneous case.

Let us make a remark about some additional amusing factorization property of the polynomials \overline{Q}_{nl} whose meaning is still to realize, namely

\overline{Q}_{3,1}(z_{1}, z_{1} + 2i|z_{3}) = \frac{1}{12}(z_{13} + i)^{2}

\overline{Q}_{4,2}(z_{1}, z_{1} + 2i|z_{3}, z_{4}) = \frac{1}{36}(z_{13} + i)^{2}(z_{14} + i)^{2}

\overline{Q}_{4,1}(z_{1}, z_{1} + 2i|z_{3}, z_{4}) = \frac{1}{18}(z_{13} + i)^{2}(z_{14} + i)^{2}

\overline{Q}_{5,2}(z_{1}, z_{1} + 2i|z_{3}, z_{4}|z_{5}) = \frac{1}{6}(z_{13} + i)^{2}(z_{14} + i)^{2}(z_{15} + i)^{2} \overline{Q}_{31}(z_{3}, z_{4}|z_{5})

\overline{Q}_{5,1}(z_{1}, z_{1} + 2i|z_{3}, z_{4}|z_{5}) = \frac{1}{24}(z_{13} + i)^{2}(z_{14} + i)^{2}(z_{15} + i)^{2}

\overline{Q}_{6,3}(z_{1}, z_{1} + 2i|z_{3}, z_{4}|z_{5}, z_{6}) = \frac{1}{6}(z_{13} + i)^{2}(z_{14} + i)^{2}(z_{15} + i)^{2}(z_{16} + i)^{2} \overline{Q}_{4,2}(z_{3}, z_{4}|z_{5}, z_{6})

\overline{Q}_{6,2}(z_{1}, z_{1} + 2i|z_{3}, z_{4}|z_{5}, z_{6}) = \frac{1}{6}(z_{13} + i)^{2}(z_{14} + i)^{2}(z_{15} + i)^{2}(z_{16} + i)^{2} \overline{Q}_{4,1}(z_{3}, z_{4}|z_{5}, z_{6})

\overline{Q}_{6,1}(z_{1}, z_{1} + 2i|z_{3}, z_{4}|z_{5}, z_{6}) = \frac{1}{30}(z_{13} + i)^{2}(z_{14} + i)^{2}(z_{15} + i)^{2}(z_{16} + i)^{2}

Looking at these formulae we can assume that

\frac{\overline{Q}_{n,l}(z_{1}, z_{1} + 2i|z_{3}, z_{4}| … |z_{2l−1}z_{2l}|z_{2l+1} … z_{n})}{\overline{Q}_{n−2,l−1}(z_{3}, z_{4}| … |z_{2l−1}z_{2l}|z_{2l+1} … z_{n})} = \frac{1}{6} ∏_{j=3}^{n}(z_{1j} + i)^{2}  (3.71)

4. The structure of P (n) in homogeneous case

As we have discussed in [10] the formulae in the homogeneous case become more regular in terms of the alternating zeta values

ζ_{a}(s) = ∑_{n>0} \frac{(−1)^{n−1}}{n^{s}} = −Li_{s}(−1)  (4.1)

where Li_{s}(x) is the polylogarithm. The alternating zeta series is related to the Riemann zeta function as follows

ζ(s) = \frac{1}{1 − 2^{1−s}} ζ_{a}(s)  (4.2)

Unlike Riemann zeta function the alternating zeta series is regular at s = 1, it is ζ_{a}(1) = log 2. Let us mention that the answers for P (1), …, P (4) can be obtained from the formulae (3.2-3.5) by taking the homogeneous limit i.e. z_{j} → 0 and using the expansion of the function G (3.13).

The same can be also done for n = 5 and n = 6 by means of the formula (3.20), formulae (3.21), (3.22), (3.23) and the answers for the polynomials (3.48-3.49) and (3.65-3.67) for the cases n = 5 and n = 6 respectively.

For n ≤ 5 we reproduce the known results, see [10] formula (1.16) in there.

In the case n = 6 we discover a new result

P (6) = \frac{1}{7}\{1 − 35 ζ_{a}(1) + 322 ζ_{a}(3) − \frac{9244}{5} ζ_{a}(5) + \frac{22694}{5} ζ_{a}(7) − 2982 ζ_{a}(9)

− \frac{3920}{3} ζ_{a}(1) · ζ_{a}(3) + \frac{369908}{15} ζ_{a}(1) · ζ_{a}(5) − \frac{28784}{5} ζ_{a}(3)^{2} − \frac{263816}{3} ζ_{a}(1) · ζ_{a}(7)

+ \frac{3458}{15} ζ_{a}(3) · ζ_{a}(5) + \frac{323344}{5} ζ_{a}(1) · ζ_{a}(9) + \frac{933702}{5} ζ_{a}(3) · ζ_{a}(7) − \frac{751592}{9} ζ_{a}(5)^{2}

− \frac{2627842}{15} ζ_{a}(3) · ζ_{a}(9) + \frac{235963}{9} ζ_{a}(5) · ζ_{a}(7) + \frac{368564}{3} ζ_{a}(5) · ζ_{a}(9) − \frac{644987}{9} ζ_{a}(7)^{2}

+ \frac{538496}{45} ζ_{a}(1) · ζ_{a}(3) · ζ_{a}(5) − \frac{269248}{135} ζ_{a}(3)^{3} − \frac{1143268}{9} ζ_{a}(1) · ζ_{a}(3) · ζ_{a}(7) + \frac{653296}{9} ζ_{a}(1) · ζ_{a}(5)^{2}

− \frac{163324}{45} ζ_{a}(3)^{2} · ζ_{a}(5) + \frac{1737148}{15} ζ_{a}(1) · ζ_{a}(3) · ζ_{a}(9) − \frac{1737148}{45} ζ_{a}(3)^{2} · ζ_{a}(7) + \frac{124082}{9} ζ_{a}(3) · ζ_{a}(5)^{2}

− \frac{528164}{3} ζ_{a}(1) · ζ_{a}(5) · ζ_{a}(9) + \frac{924287}{9} ζ_{a}(1) · ζ_{a}(7)^{2} + \frac{264082}{5} ζ_{a}(3)^{2} · ζ_{a}(9)

− \frac{264082}{9} ζ_{a}(3) · ζ_{a}(5) · ζ_{a}(7) + \frac{188630}{27} ζ_{a}(5)^{3}\}  (4.3)

This support our hypothesis that all P (n) can be expression in terms of values of Riemann zeta functions at odd arguments, log 2 and rational coefficients.

From this expression we can get numerical value

P (6) = 7.0685928 · 10^{−9}  (4.4)

In [10] a numerical method was used for evaluation of P (6). It is called Density Matrix Renormalization Group (DMRG). The results can be found in Table 1 in [10]. In particular P (6) = 7.05 · 10^{−9} with an uncertainty in the second digit after the decimal point. It is in a good agreement with our analytic result (4.3), (4.4).

Looking at the formulae (1.16) of [10] and the above expression (4.3) we can make a general conjecture for the dependence of P (n) on the alternating zeta series

P (n) = \frac{1}{(n + 1)} ∑_{r⃗∈U} B_{r_{0},r_{1},...,r_{n−2}}^{(n)} ∏_{j=0}^{n−2} [ζ_{a}(2 j + 1)]^{r_{j}}  (4.5)

where all coefficients B_{r_{0},r_{1},...,r_{n−2}}^{(n)} are rational and the sum is over non-negative integers r_{0}, …, r_{n−2} which belong to the region U determined by the following two conditions

∑_{j=0}^{n−2} r_{j} ≤ [n/2]

∑_{j=0}^{n−2} r_{j} (2j + 1) ≤ \frac{n(n − 1)}{2}  (4.6)

Let us show how the non-zero coefficients B_{r_{0},r_{1},...,r_{n−2}}^{(n)} look like for the cases when we know the manifest analytic answer, namely, when n = 1, 2, … 6

B^{(1)} = 1
B_{0}^{(2)} = 1 B_{1}^{(2)} = −1
B_{0,0}^{(3)} = 1 B_{1,0}^{(3)} = −4 B_{0,1}^{(3)} = 2

B_{0,0,0}^{(4)} = 1 B_{1,0,0}^{(4)} = −10 B_{0,1,0}^{(4)} = \frac{173}{9} B_{0,0,1}^{(4)} = −\frac{110}{9}

B_{1,1,0}^{(4)} = −\frac{110}{9} B_{1,0,1}^{(4)} = \frac{170}{9} B_{0,2,0}^{(4)} = −\frac{17}{3}

B_{0,0,0,0}^{(5)} = 1 B_{1,0,0,0}^{(5)} = −20 B_{0,1,0,0}^{(5)} = \frac{281}{3} B_{0,0,1,0}^{(5)} = −\frac{1355}{6} B_{0,0,0,1}^{(5)} = \frac{889}{6}

B_{1,1,0,0}^{(5)} = −180 B_{1,0,1,0}^{(5)} = \frac{3920}{3} B_{1,0,0,1}^{(5)} = −\frac{3290}{3} B_{0,1,1,0}^{(5)} = −\frac{170}{3} B_{0,1,0,1}^{(5)} = 679

B_{0,2,0,0}^{(5)} = −326 B_{0,0,2,0}^{(5)} = −\frac{970}{3}

B_{0,0,0,0,0}^{(6)} = 1

B_{1,0,0,0,0}^{(6)} = −35 B_{0,1,0,0,0}^{(6)} = 322 B_{0,0,1,0,0}^{(6)} = −\frac{9244}{5} B_{0,0,0,1,0}^{(6)} = \frac{22694}{5} B_{0,0,0,0,1}^{(6)} = −2982

B_{1,1,0,0,0}^{(6)} = −\frac{3920}{3} B_{1,0,1,0,0}^{(6)} = \frac{369908}{15} B_{1,0,0,1,0}^{(6)} = −\frac{263816}{3} B_{1,0,0,0,1}^{(6)} = \frac{323344}{5}

B_{0,1,1,0,0}^{(6)} = \frac{3458}{15} B_{0,1,0,1,0}^{(6)} = \frac{933702}{5} B_{0,1,0,0,1}^{(6)} = −\frac{2627842}{15} B_{0,0,1,1,0}^{(6)} = \frac{235963}{9}

B_{0,0,1,0,1}^{(6)} = \frac{368564}{3} B_{0,2,0,0,0}^{(6)} = −\frac{28784}{5} B_{0,0,2,0,0}^{(6)} = −\frac{751592}{9} B_{0,0,0,2,0}^{(6)} = −\frac{644987}{9}

B_{1,1,1,0,0}^{(6)} = \frac{538496}{45} B_{1,1,0,1,0}^{(6)} = −\frac{1143268}{9} B_{1,1,0,0,1}^{(6)} = \frac{1737148}{15} B_{1,0,1,0,1}^{(6)} = −\frac{528164}{3}

B_{0,1,1,1,0}^{(6)} = −\frac{264082}{9} B_{1,0,2,0,0}^{(6)} = \frac{653296}{9} B_{1,0,0,2,0}^{(6)} = \frac{924287}{9} B_{0,1,2,0,0}^{(6)} = \frac{124082}{9}

B_{0,2,1,0,0}^{(6)} = −\frac{163324}{45} B_{0,2,0,1,0}^{(6)} = −\frac{1737148}{45} B_{0,2,0,0,1}^{(6)} = \frac{264082}{5}

B_{0,3,0,0,0}^{(6)} = −\frac{269248}{135} B_{0,0,3,0,0}^{(6)} = \frac{188630}{27}  (4.7)

For us it was a bit surprising that two of coefficients B^{(6)} appeared to be zero, namely,

B_{1,0,1,1,0}^{(6)} = 0 B_{1,2,0,0,0}^{(6)} = 0  (4.8)

It means that the structures ζ_{a}(1) · ζ_{a}(5) · ζ_{a}(7) and ζ_{a}(1) · ζ_{a}(3)^{2} do not appear in the final answer for P (6). Meanwhile the term ζ_{a}(3)^{3} survived with the non-zero coefficient B_{0,3,0,0,0}^{(6)} which we expected to be zero.

The most evident conjecture that we can make looking at the formulae (4.7) is as follows

B_{0,0,...,0}^{(n)} = 1,  (4.9)
B_{1,0,...,0}^{(n)} = − ( \begin{matrix}n + 1 \\ 3 \end{matrix} )  (4.10)

where ( \begin{matrix}n \\ m \end{matrix} ) is the binomial coefficient.

The next our conjecture is less trivial. As appeared the coefficients B might satisfy some equations. One of them has the following form

∑_{r_{0}+r_{1}+...+r_{n−2} = p} B_{r_{0},r_{1},...,r_{n−2}}^{(n)} = (−1)^{p} ( \begin{matrix}n − p \\ p \end{matrix} )  (4.11)

where p is some fixed positive integer. The expression (4.11) can be easily verified for the first coefficients given by the formulae (4.7). For example, when p = 1 we get the following equation

B_{1,0,...,0}^{(n)} + B_{0,1,...,0}^{(n)} + … + B_{0,0,...,1}^{(n)} = − n + 1  (4.12)

where we have already made the conjecture (4.10) for the first term. We believe that there should be more equations like (4.11) which probably provide the rigorous expression for the coefficients B.

We have pointed out in our previous work [10] that since ζ_{a}(1), ζ_{a}(3), ζ_{a}(5), … are very likely different irrational (or even transcendental) numbers ^{5} P (n) seem to be different irrational (or transcendental) numbers as well. This means that P (n) does not satisfy polynomial recursion relation with respect to the distance.

In order to clarify the structure of the formula (4.5) let us formally replace all ζ_{a}(2j + 1) by one complex variable x. This will define a new function P (n, x).

Then we can calculate P (n, x) using our conjectures (4.5), (4.11) and some properties of the binomial coefficients. Namely,

P (n, x) = ∑_{p=0}^{[n/2]}(−x)^{p} ( \begin{matrix}n − p \\ p \end{matrix} ) = \frac{A_{+}^{n+1} − A_{−}^{n+1}}{(n + 1)\sqrt{1 − 4x}}  (4.13)

5It was proven by Apéry [17] that ζ(3) is irrational. Then Rivoal [19] proved that one of the nine numbers ζ(5), …, ζ(21) is irrational. One of the most recent theorem proved by Zudilin [20] says that one of the four values ζ(5), …, ζ(11) is irrational. (See also the paper by D. Zagier [18] and the paper by Yu. Nesterenko [21]).

where

A_{±} = \frac{1 ± \sqrt{1 − 4x}}{2}  (4.14)

In particular when x → 1 one gets

P (n, 1) = \{ \begin{matrix}1, n = 6k or n = 6k + 1 \\ 0, n = 3k + 2 \\ −1, n = 6k + 3 or 6k + 4 \end{matrix}  (4.15)

Note that the alternating zeta values approaches 1 as the argument gos to infinity

lim_{s→∞} ζ_{a}(s) = 1  (4.16)

Another nice form of the formula (4.13) can be obtained by substitution

x = \frac{1}{4 cosh^{2} α}  (4.17)

Then

P (n, \frac{1}{4 cosh^{2} α}) = \frac{sinh [(n + 1)α]}{2^{n}(n + 1) cosh^{n} α sinh α}  (4.18)

We see that when α tends to zero

lim_{α→0} P (n, \frac{1}{4 cosh^{2} α}) = \frac{1}{2^{n}}  (4.19)

Let us remark that this result appeared to coincide with the limiting formula (3.26). We do not know if this is accidental or there is a reason for this.

Let us briefly discuss the generating function for the values P (n)

Ψ(y) = ∑_{n=0}^{∞} y^{n} P (n)  (4.20)

where P (0) = 1 by definition. Taking into account the conjectures (4.9) and (4.10) we can easily get the two first terms for the generating function Ψ(y)

Ψ(y) = −\frac{ln (1 − y)}{y} + \frac{y^{2}}{3(1 − y)^{3}} ln 2 + …  (4.21)

As we discussed in [8] and [10] we expect that for n ≫ 1

P (n) ∼ e^{−κn^{2}}  (4.22)

If we substitute it formally into eq. (4.20) with y = e^{u} then we can expect that

Ψ(e^{u}) ∼ Ψ̃(u) = ∑_{n=0}^{∞} e^{−κn^{2}+u n}  (4.23)

so that the function Ψ̃(u) satisfies the functional equation

Ψ̃(u) + Ψ̃(−u) − 1 = θ_{3}(\frac{iu}{2}, e^{−κ})  (4.24)

where θ_{3} is the third Jacobi theta function with the nome e^{−κ}. So we can expect that the generating function (4.20) may be related with the elliptic functions and have some automorphic properties.

Another possibility is to put into the r.h.s. of eq. (4.20) the values P (n, x) given by (4.13) instead of P (n) then

Ψ(x, y) = ∑_{n=0}^{∞} y^{n} P (n, x) = \frac{1}{y\sqrt{1 − 4x}} ln \frac{2 − y + y\sqrt{1 − 4x}}{2 − y − y\sqrt{1 − 4x}}  (4.25)

Another generating function looks much simpler

Ψ′(x, y) = ∑_{n=0}^{∞}(n + 1)y^{n} P (n, x) = \frac{1}{1 − y + xy^{2}}  (4.26)

5. Discussion

The main point of this communication was to consider the emptiness formation probability in the inhomogeneous case. The basic advantage of the inhomogeneous case is that we have more parameters at our disposal and relation to the qKZ. We derived three general properties of P_{n} from qKZ, which appeared to be extremely useful. From our experience of the direct calculation of P_{n} with n ≤ 4 we conjectured a general ansatz for P_{n} (3.20). We established that the ansatz (3.20) with the first and the second property from the Section 2 [ namely (2.8)] completely fix the answer. The third property (2.9) turns out to be a corollary. This observation allowed us to evaluate P_{5} and P_{6} very efficiently [ in the inhomogeneous case]. The homogeneous limit of P_{5} reproduced the expression, which we obtained in paper [10] by going through hard and long computations. That time we were not sure that we will be able to calculate P_{6} at all. Now it is possible to do it very quickly just by taking the homogeneous limit of our result in the inhomogeneous case. In the next publications we are planning to prove our general ansatz (3.20) and evaluate P (n) for arbitrary n. Actually we think that qKZ approach is so powerful that we will be able to evaluate any correlation function in the Heisenberg XXX model and to show that it has a structure similar to P (n)

F (z_{1}, …, z_{n})_{ϵ_{1},...,ϵ_{n}} = ∑_{l=0}^{[ \frac{n}{2} ]}\{A_{n,l}^{ϵ_{1},...,ϵ_{n}} ∏_{j=1}^{l} G(z_{2j−1} − z_{2j}) + permutations\}  (5.1)

where ϵ_{j} = ±1

F (z_{1}, …, z_{n})_{ϵ_{1},...,ϵ_{n}} = ⟨GS| ∏_{j=1}^{n} P_{j}^{ϵ_{j}}|GS⟩,  (5.2)

and

P_{j}^{±} = \frac{1 ± σ_{j}^{z}}{2}

We may also expect that similar to the functions A_{n,l} = A_{n,l}^{+,...,+} from (3.20) the functions A_{n,l}^{ϵ_{1},...,ϵ_{n}} are rational of their arguments z_{1}, …, z_{n} also.

6. Acknowledgements

The authors would like to thank A. Abanov, R. Flume, M. Jimbo, D. Kreimer, M. Lashkevich, S. Lukyanov, T. Miwa, Yu. Nesterenko, P. Pyatov, V. Tarasov, D. Zagier and A. Zamolodchikov for useful discussions. This research has been supported by the following grants: NSF grant PHY-9988566, the Russian Foundation of Basic Research under grant # 01–01–00201, by INTAS under grants #00-00055 and # 00-00561. HEB would like to thank the administration of the Max-Planck Institute for Mathematics for hospitality and perfect conditions for the work. HEB and FAS are grateful to organizers of the International Workshop “Conformal Field Theory and Integrable Models”, Chernogolovka, September 2002, Russia for opportunity to present this work.

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