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Abdelaziz, Y.; Boukraa, S.; Koutschan, C.; Maillard, J.-M.

Diagonals of rational functions, pullbacked \(\mathbf{_2F_1}\) hypergeometric functions and modular forms. (English) Zbl 1408.33008

J. Phys. A, Math. Theor. 51, No. 45, Article ID 455201, 30 p. (2018).
Summary: We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that the diagonal of a seven parameter rational function of three variables with a numerator equal to one and a denominator which is a polynomial of degree at most two, can be expressed as a pullbacked \({}_2F_1\) hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked \({}_2F_1\) hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitly that the diagonal is a modular form. We then generalize this result to nine and ten parameter families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We show that each of these rational functions yields an infinite number of rational functions whose diagonals are also pullbacked \({}_2F_1\) hypergeometric functions and modular forms.

Cited in 4 Documents

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
11F03 Modular and automorphic functions

Keywords:

diagonals of rational functions; pullbacked hypergeometric functions; modular forms; modular equations; hauptmoduls; creative telescoping; series with integer coefficients
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References:

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