## Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms

Youssef Abdelaziz, Salah Boukraa, Christoph Koutschan, Jean-Marie Maillard

### Abstract

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 2F1 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 2F1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely 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 2F1 hypergeometric functions and modular forms.

### Paper

The full paper is available here.

### Supplementary electronic material

#### Seven-parameter rational function

In Sections 2.2 – 2.4 we study a seven-parameter family of rational functions given in Equation (7). Its diagonal can be represented as a 2F1 hypergeometric function involving the following two polynomials P2(x) and P4(x), given in Equations (16) and (17), respectively. Here we give these two polynomials in a computer-readable format:

 P2(x) = 24*a*c1*c2*c3*x^2 + 16*b1^2*c1^2*x^2 - 16*b1*b2*c1*c2*x^2 - 16*b1*b3*c1*c3*x^2 + 16*b2^2*c2^2*x^2 - 16*b2*b3*c2*c3*x^2 + 16*b3^2*c3^2*x^2 - 8*a^2*b1*c1*x - 8*a^2*b2*c2*x - 8*a^2*b3*c3*x + 24*a*b1*b2*b3*x + a^4 P4(x) = 216*c1^2*c2^2*c3^2*x^4 - 144*a*b1*c1^2*c2*c3*x^3 - 144*a*b2*c1*c2^2*c3*x^3 - 144*a*b3*c1*c2*c3^2*x^3 - 64*b1^3*c1^3*x^3 + 96*b1^2*b2*c1^2*c2*x^3 + 96*b1^2*b3*c1^2*c3*x^3 + 96*b1*b2^2*c1*c2^2*x^3 + 48*b1*b2*b3*c1*c2*c3*x^3 + 96*b1*b3^2*c1*c3^2*x^3 - 64*b2^3*c2^3*x^3 + 96*b2^2*b3*c2^2*c3*x^3 + 96*b2*b3^2*c2*c3^2*x^3 - 64*b3^3*c3^3*x^3 + 36*a^3*c1*c2*c3*x^2 + 48*a^2*b1^2*c1^2*x^2 + 24*a^2*b1*b2*c1*c2*x^2 + 24*a^2*b1*b3*c1*c3*x^2 + 48*a^2*b2^2*c2^2*x^2 + 24*a^2*b2*b3*c2*c3*x^2 + 48*a^2*b3^2*c3^2*x^2 - 144*a*b1^2*b2*b3*c1*x^2 - 144*a*b1*b2^2*b3*c2*x^2 - 144*a*b1*b2*b3^2*c3*x^2 + 216*b1^2*b2^2*b3^2*x^2 - 12*a^4*b1*c1*x - 12*a^4*b2*c2*x - 12*a^4*b3*c3*x + 36*a^3*b1*b2*b3*x + a^6

#### Nine-parameter rational function

In Section 3.1 we study a nine-parameter family of rational functions given in Equation (59). Its diagonal can be represented as a 2F1 hypergeometric function involving the following two polynomials P4(x) and P6(x), given in Equations (61) and (62), respectively. Here we give these two polynomials in a computer-readable format:

 P4(x) = 16*d^2*e^2*x^4 - 16*b1*c1*d*e*x^3 + 224*b2*c2*d*e*x^3 - 16*b3*c3*d*e*x^3 - 48*c1^2*c2*d*x^3 - 48*c2*c3^2*e*x^3 - 8*a^2*d*e*x^2 + 24*a*b1*c3*e*x^2 + 24*a*b3*c1*d*x^2 + 24*a*c1*c2*c3*x^2 - 48*b1^2*b2*e*x^2 + 16*b1^2*c1^2*x^2 - 16*b1*b2*c1*c2*x^2 - 16*b1*b3*c1*c3*x^2 + 16*b2^2*c2^2*x^2 - 48*b2*b3^2*d*x^2 - 16*b2*b3*c2*c3*x^2 + 16*b3^2*c3^2*x^2 - 8*a^2*b1*c1*x - 8*a^2*b2*c2*x - 8*a^2*b3*c3*x + 24*a*b1*b2*b3*x + a^4 P6(x) = -64*d^3*e^3*x^6 + 96*b1*c1*d^2*e^2*x^5 + 2112*b2*c2*d^2*e^2*x^5 + 96*b3*c3*d^2*e^2*x^5 - 576*c1^2*c2*d^2*e*x^5 - 576*c2*c3^2*d*e^2*x^5 + 48*a^2*d^2*e^2*x^4 - 144*a*b1*c3*d*e^2*x^4 - 144*a*b3*c1*d^2*e*x^4 + 720*a*c1*c2*c3*d*e*x^4 - 576*b1^2*b2*d*e^2*x^4 + 96*b1^2*c1^2*d*e*x^4 + 216*b1^2*c3^2*e^2*x^4 - 960*b1*b2*c1*c2*d*e*x^4 + 48*b1*b3*c1*c3*d*e*x^4 + 288*b1*c1^3*c2*d*x^4 - 144*b1*c1*c2*c3^2*e*x^4 + 2112*b2^2*c2^2*d*e*x^4 - 576*b2*b3^2*d^2*e*x^4 - 960*b2*b3*c2*c3*d*e*x^4 - 576*b2*c1^2*c2^2*d*x^4 - 576*b2*c2^2*c3^2*e*x^4 + 216*b3^2*c1^2*d^2*x^4 + 96*b3^2*c3^2*d*e*x^4 - 144*b3*c1^2*c2*c3*d*x^4 + 288*b3*c2*c3^3*e*x^4 + 216*c1^2*c2^2*c3^2*x^4 + 24*a^2*b1*c1*d*e*x^3 - 480*a^2*b2*c2*d*e*x^3 + 24*a^2*b3*c3*d*e*x^3 - 72*a^2*c1^2*c2*d*x^3 - 72*a^2*c2*c3^2*e*x^3 - 144*a*b1^2*c1*c3*e*x^3 + 720*a*b1*b2*b3*d*e*x^3 + 720*a*b1*b2*c2*c3*e*x^3 - 144*a*b1*b3*c1^2*d*x^3 - 144*a*b1*b3*c3^2*e*x^3 - 144*a*b1*c1^2*c2*c3*x^3 + 720*a*b2*b3*c1*c2*d*x^3 - 144*a*b2*c1*c2^2*c3*x^3 - 144*a*b3^2*c1*c3*d*x^3 - 144*a*b3*c1*c2*c3^2*x^3 + 288*b1^3*b2*c1*e*x^3 - 64*b1^3*c1^3*x^3 - 576*b1^2*b2^2*c2*e*x^3 - 144*b1^2*b2*b3*c3*e*x^3 + 96*b1^2*b2*c1^2*c2*x^3 + 96*b1^2*b3*c1^2*c3*x^3 + 96*b1*b2^2*c1*c2^2*x^3 - 144*b1*b2*b3^2*c1*d*x^3 + 48*b1*b2*b3*c1*c2*c3*x^3 + 96*b1*b3^2*c1*c3^2*x^3 - 64*b2^3*c2^3*x^3 - 576*b2^2*b3^2*c2*d*x^3 + 96*b2^2*b3*c2^2*c3*x^3 + 288*b2*b3^3*c3*d*x^3 + 96*b2*b3^2*c2*c3^2*x^3 - 64*b3^3*c3^3*x^3 - 12*a^4*d*e*x^2 + 36*a^3*b1*c3*e*x^2 + 36*a^3*b3*c1*d*x^2 + 36*a^3*c1*c2*c3*x^2 - 72*a^2*b1^2*b2*e*x^2 + 48*a^2*b1^2*c1^2*x^2 + 24*a^2*b1*b2*c1*c2*x^2 + 24*a^2*b1*b3*c1*c3*x^2 + 48*a^2*b2^2*c2^2*x^2 - 72*a^2*b2*b3^2*d*x^2 + 24*a^2*b2*b3*c2*c3*x^2 + 48*a^2*b3^2*c3^2*x^2 - 144*a*b1^2*b2*b3*c1*x^2 - 144*a*b1*b2^2*b3*c2*x^2 - 144*a*b1*b2*b3^2*c3*x^2 + 216*b1^2*b2^2*b3^2*x^2 - 12*a^4*b1*c1*x - 12*a^4*b2*c2*x - 12*a^4*b3*c3*x + 36*a^3*b1*b2*b3*x + a^6

We also provide a Maple Worksheet and a Mathematica notebook where these two polynomials are defined and where the correctness of our result is verified empirically.

#### Ten-parameter rational function

In Section 3.2 we study a ten-parameter family of rational functions given in Equation (63). Its diagonal can be represented as a 2F1 hypergeometric function involving the following two polynomials P3(x) and P6(x), given in Equation (66) and in Appendix D, respectively. Here we give these two polynomials in a computer-readable format:

 P3(x) = -216*a*d1*d2*d3*x^3 + 144*b1*c3*d2*d3*x^3 + 144*b2*c1*d1*d3*x^3 + 144*b3*c2*d1*d2*x^3 - 48*c1^2*c2*d1*x^3 - 48*c1*c3^2*d3*x^3 - 48*c2^2*c3*d2*x^3 + 24*a*b1*c2*d2*x^2 + 24*a*b2*c3*d3*x^2 + 24*a*b3*c1*d1*x^2 + 24*a*c1*c2*c3*x^2 - 48*b1^2*b3*d2*x^2 + 16*b1^2*c1^2*x^2 - 48*b1*b2^2*d3*x^2 - 16*b1*b2*c1*c2*x^2 - 16*b1*b3*c1*c3*x^2 + 16*b2^2*c2^2*x^2 - 48*b2*b3^2*d1*x^2 - 16*b2*b3*c2*c3*x^2 + 16*b3^2*c3^2*x^2 - 8*a^2*b1*c1*x - 8*a^2*b2*c2*x - 8*a^2*b3*c3*x + 24*a*b1*b2*b3*x + a^4 P6(x) = -5832*d1^2*d2^2*d3^2*x^6 + 3888*b1*c2*d1*d2^2*d3*x^5 + 3888*b2*c3*d1*d2*d3^2*x^5 + 3888*b3*c1*d1^2*d2*d3*x^5 - 864*c1^3*d1^2*d3*x^5 - 1296*c1*c2*c3*d1*d2*d3*x^5 - 864*c2^3*d1*d2^2*x^5 - 864*c3^3*d2*d3^2*x^5 - 1296*a*b1*c1*d1*d2*d3*x^4 - 1296*a*b2*c2*d1*d2*d3*x^4 - 1296*a*b3*c3*d1*d2*d3*x^4 + 864*a*c1^2*c3*d1*d3*x^4 + 864*a*c1*c2^2*d1*d2*x^4 + 864*a*c2*c3^2*d2*d3*x^4 - 864*b1^3*d2^2*d3*x^4 + 864*b1^2*c1*c3*d2*d3*x^4 + 216*b1^2*c2^2*d2^2*x^4 - 1296*b1*b2*b3*d1*d2*d3*x^4 + 864*b1*b2*c1^2*d1*d3*x^4 - 1296*b1*b2*c2*c3*d2*d3*x^4 - 1296*b1*b3*c1*c2*d1*d2*x^4 + 864*b1*b3*c3^2*d2*d3*x^4 + 288*b1*c1^3*c2*d1*x^4 - 576*b1*c1^2*c3^2*d3*x^4 - 144*b1*c1*c2^2*c3*d2*x^4 - 864*b2^3*d1*d3^2*x^4 + 864*b2^2*c1*c2*d1*d3*x^4 + 216*b2^2*c3^2*d3^2*x^4 - 1296*b2*b3*c1*c3*d1*d3*x^4 + 864*b2*b3*c2^2*d1*d2*x^4 - 576*b2*c1^2*c2^2*d1*x^4 - 144*b2*c1*c2*c3^2*d3*x^4 + 288*b2*c2^3*c3*d2*x^4 - 864*b3^3*d1^2*d2*x^4 + 216*b3^2*c1^2*d1^2*x^4 + 864*b3^2*c2*c3*d1*d2*x^4 - 144*b3*c1^2*c2*c3*d1*x^4 + 288*b3*c1*c3^3*d3*x^4 - 576*b3*c2^2*c3^2*d2*x^4 + 216*c1^2*c2^2*c3^2*x^4 + 540*a^3*d1*d2*d3*x^3 - 648*a^2*b1*c3*d2*d3*x^3 - 648*a^2*b2*c1*d1*d3*x^3 - 648*a^2*b3*c2*d1*d2*x^3 - 72*a^2*c1^2*c2*d1*x^3 - 72*a^2*c1*c3^2*d3*x^3 - 72*a^2*c2^2*c3*d2*x^3 + 864*a*b1^2*b2*d2*d3*x^3 - 144*a*b1^2*c1*c2*d2*x^3 + 720*a*b1*b2*c1*c3*d3*x^3 - 144*a*b1*b2*c2^2*d2*x^3 + 864*a*b1*b3^2*d1*d2*x^3 - 144*a*b1*b3*c1^2*d1*x^3 + 720*a*b1*b3*c2*c3*d2*x^3 - 144*a*b1*c1^2*c2*c3*x^3 + 864*a*b2^2*b3*d1*d3*x^3 - 144*a*b2^2*c2*c3*d3*x^3 + 720*a*b2*b3*c1*c2*d1*x^3 - 144*a*b2*b3*c3^2*d3*x^3 - 144*a*b2*c1*c2^2*c3*x^3 - 144*a*b3^2*c1*c3*d1*x^3 - 144*a*b3*c1*c2*c3^2*x^3 + 288*b1^3*b3*c1*d2*x^3 - 64*b1^3*c1^3*x^3 - 576*b1^2*b2^2*c1*d3*x^3 - 144*b1^2*b2*b3*c2*d2*x^3 + 96*b1^2*b2*c1^2*c2*x^3 - 576*b1^2*b3^2*c3*d2*x^3 + 96*b1^2*b3*c1^2*c3*x^3 + 288*b1*b2^3*c2*d3*x^3 - 144*b1*b2^2*b3*c3*d3*x^3 + 96*b1*b2^2*c1*c2^2*x^3 - 144*b1*b2*b3^2*c1*d1*x^3 + 48*b1*b2*b3*c1*c2*c3*x^3 + 96*b1*b3^2*c1*c3^2*x^3 - 64*b2^3*c2^3*x^3 - 576*b2^2*b3^2*c2*d1*x^3 + 96*b2^2*b3*c2^2*c3*x^3 + 288*b2*b3^3*c3*d1*x^3 + 96*b2*b3^2*c2*c3^2*x^3 - 64*b3^3*c3^3*x^3 + 36*a^3*b1*c2*d2*x^2 + 36*a^3*b2*c3*d3*x^2 + 36*a^3*b3*c1*d1*x^2 + 36*a^3*c1*c2*c3*x^2 - 72*a^2*b1^2*b3*d2*x^2 + 48*a^2*b1^2*c1^2*x^2 - 72*a^2*b1*b2^2*d3*x^2 + 24*a^2*b1*b2*c1*c2*x^2 + 24*a^2*b1*b3*c1*c3*x^2 + 48*a^2*b2^2*c2^2*x^2 - 72*a^2*b2*b3^2*d1*x^2 + 24*a^2*b2*b3*c2*c3*x^2 + 48*a^2*b3^2*c3^2*x^2 - 144*a*b1^2*b2*b3*c1*x^2 - 144*a*b1*b2^2*b3*c2*x^2 - 144*a*b1*b2*b3^2*c3*x^2 + 216*b1^2*b2^2*b3^2*x^2 - 12*a^4*b1*c1*x - 12*a^4*b2*c2*x - 12*a^4*b3*c3*x + 36*a^3*b1*b2*b3*x + a^6

We also provide a Maple Worksheet and a Mathematica notebook where these two polynomials are defined and where the correctness of our result is verified empirically.

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