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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics
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Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics

机译：晶格统计力学和枚举组合学中的非完整函数

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We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the nonlinear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form 2 + 2 cos(j pi), the holonomic status of q = 4 being unclear. We focus on the analysis of the q = 4 case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this q = 4 series modulo the first eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question of whether such remarkable non-holonomic functions can be seen as a ratio of diagonals of rational functions, or even algebraic functions of diagonals of rational functions.
机译：我们记得伊辛模型的完整磁化率级数，素数2的模幂归结为代数函数。我们还记得由Tutte获得的非线性多项式微分方程，用于通过顶点生成q色根三角剖分的生成函数，已知该函数对于2 + 2 cos（j pi / n）形式的所有数字都有代数解， q = 4的完整状态尚不清楚。我们专注于q = 4情况的分析，表明相应的序列肯定是非完整的。沿着先前关于Ising模型的敏感性的工作，我们认为此q = 4级数是对前八个素数2、3，... 19取模的模，并表明该（可能是非完整的）函数按模减小这些素数，代数函数。我们猜想，这个可能是非完整的函数会减少（几乎）每个质数或质数幂的模代数函数。这就提出了一个问题，即这种显着的非完整函数可以看作是有理函数的对角线的比率，甚至可以看作有理函数的对角线的代数函数。

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