Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions

Abstract: In present paper, with the help of the Faà di Bruno formula and identities of partial Bell polynomials, the author establishes explicit formulas of the Gauss hypergeometric functions \begin{gather*} {\,}2F_1\biggl(\frac{1-n}{2},\frac{2-n}{2};\frac{3}{2}-m;z^2\biggr), \quad {\,}_2F_1\biggl(-\frac{n}{2},\frac{1-n}{2};\frac{1}{2}-m;z^2\biggr),\ {\,}_2F_1\biggl(a,a+\frac{1}{2};\frac{3}{2}-m;z^2\biggr), \quad {\,}_2F_1\biggl(a,a+\frac{1}{2};\frac{1}{2}-m;z^2\biggr) \end{gather*} for $m,n\in\mathbb{N}$ and $a\in\mathbb{C}$, and then derives two combinatorial identities \begin{equation*} \sum{k=0}^{m}\frac{2^k}{k!} \binom{2m-2k}{m-k} \sum_{\ell=0}^{k} \frac{(-1)^\ell}{2^\ell} \frac{(2k-2\ell-1)!!}{(n-\ell)!} \binom{2k-\ell-1}{\ell-1} =\frac{1}{n!}\binom{2m-n}{m} \end{equation*} and \begin{equation*} \sum_{k=1}^{m}\frac{1}{(k!)^2}\binom{2m-2k}{m-k} \sum_{\ell=1}^{k} \binom{k}{\ell}\ell(2k-\ell-1)! (2a)\ell =\binom{2m+2a}{m}, \end{equation*} where $m\in\mathbb{N}_0$, $n\in\mathbb{Z}$, and $a\in\mathbb{C}$. These newly-established identities generalize the nice and beautiful combinatorial identity \begin{equation*} \sum{k=0}^{n} \frac{2^{k}}{k!}\binom{2n-2k}{n-k} \sum_{j=0}^{k}\frac{(-1)^{j}}{2^j} \frac{(2k-2j-1)!!}{(n-j)!} \binom{2k-j-1}{j-1} =\frac{1}{n!}, \quad n\in\mathbb{N}_0, \end{equation*} which was obtained in Theorem 4 of the paper "F. Qi, C.-Y. He, and D. Lim, Explicit formulas of two Gauss hypergeometric functions and several combinatorial identities, Discrete Appl. Math., Vol. 393 (2026), 215--229. DOI: https://doi.org/10.1016/j.dam.2026.06.023".

• The paper introduces new explicit combinatorial identities derived from closed forms of Gauss hypergeometric functions using Bell polynomial techniques.
• It employs Faà di Bruno’s formula and analytic continuation to generalize classical binomial and hypergeometric identities to complex parameters.
• The approach connects hypergeometric functions with classical orthogonal polynomials, providing advanced tools for symbolic computation and combinatorial analysis.

Combinatorial Identities from Explicit Formulas of Gauss Hypergeometric Functions

Introduction

The paper "Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions" (2607.10643) investigates new connections between special function theory—specifically, explicit hypergeometric expressions—and combinatorial identities. The central methodological tool is the interplay between Faà di Bruno’s formula, partial Bell polynomials, and classical binomial identities, leading to nontrivial closed forms for hypergeometric functions and, consequently, explicit combinatorial summations. The results notably generalize previous work on hypergeometric-polynomial connections and provide new identities involving binomials, Pochhammer symbols, and double factorials.

Preliminaries: Hypergeometric Functions and Bell Polynomials

The Gauss hypergeometric function ${}_2F_1(a, b; c; z)$, ubiquitous in special function theory, is given by the series representation

$${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$

with conditions for convergence and algebraic reduction to polynomials when either $a$ or $b$ is a nonpositive integer. The explicit combinatorial content of ${}_2F_1$ functions with carefully chosen parameters emerges when exploring special selections, such as $a = -(n/2)$ or $a = (1-n)/2$, with the argument $z^2$.

The analysis leverages partial Bell polynomials $\text{B}_{n, k}$, which provide systematic expressions for derivatives of function compositions through Faà di Bruno’s formula:

$$[f \circ h(z)]^{(n)} = \sum_{k=0}^{n} f^{(k)}(h(z)) \, \text{B}_{n,k}(h'(z), h''(z), \dots, h^{(n-k+1)}(z))$$

Key identities involving specialized arguments to Bell polynomials yield closed forms in terms of double factorials and binomials, forming the backbone of later combinatorial expansions.

Explicit Formulas for Hypergeometric Polynomials

The principal technical results derive explicit formulas for four classes of hypergeometric polynomials of the form

$${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$0

and their generalizations for arbitrary complex $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$1.

Using the binomial interpretation of Pochhammer symbols and systematic application of hypergeometric differential identities (notably, the derivative formula for $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$2), along with Bell polynomial expansions for derivatives of functions of the form $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$3, the paper derives explicit expansions:

• For positive integer parameters, the hypergeometric function is written as combinations of binomial coefficients, double factorials, powers of $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$4, and alternating sign terms.
• All formulas admit direct substitutions of parameters to produce concrete polynomial identities or to connect with classical special functions (e.g., Legendre polynomials).

Main Combinatorial Identities

Two central families of combinatorial identities emerge from specialization and algebraic manipulation of the explicit hypergeometric expressions.

First Family

For $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$5, $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$6, the identity

$${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$7

is shown, generalizing earlier identities and subsuming the case $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$8 known from previous work. Notably, the result persists for $${}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}$$9, a claim proven via analytic continuation arguments employing the holomorphicity of the binomial and Pochhammer functions in $a$0.

Second Family

For $a$1 and $a$2,

$a$3

yields further combinatorial generalization, recovered through specialization in the above hypergeometric polynomial classes.

Both identities resolve, through technical manipulations, open questions posed in previous literature regarding parameter evolution and generalization beyond integral indices.

Analytical Techniques and Derivation Approach

The derivation involves heavy use of:

• Faà di Bruno's formula for higher-order derivatives of composite functions, expressed via partial Bell polynomials.
• Various closed forms for Bell polynomials evaluated at specific tuples, including those parametrized by falling and rising factorials.
• Chu–Vandermonde and related classical hypergeometric summations, particularly to evaluate convolution sums encountered in the expansion of binomially-generated summands.
• Systematic parameter substitution and analytic continuation to rigorously justify extension of combinatorial identities to complex and negative integer parameters, reliant on holomorphicity and function uniqueness theorems.

Implications and Potential Future Directions

The identities and methods provided have several implications:

• Combinatorics: The results significantly expand the landscape of binomial and hypergeometric summations with direct algorithmic consequences for evaluating sums involving non-elementary combinatorial sequences.
• Special Functions: These explicit forms make possible closed-form evaluations for wide families of hypergeometric polynomials and, as noted, connect to classical polynomials such as Legendre, Jacobi, and Gegenbauer polynomials, suggesting further investigation of their combinatorial expansions.
• Symbolic Computation: The approach provides templates for symbolic summation algorithms, particularly those using Bell polynomials for function composition derivatives.
• Analyticity as a Generalization Tool: The analytic continuation arguments employed for combinatorial identities indicate broader possibilities for extending discrete-parameter results to the complex domain.

The paper suggests that these techniques can be extended to derive more identities, specifically for classical orthogonal polynomials with hypergeometric representations and potentially for multivariate generalizations, hinting at further research in analytic combinatorics and computational special function theory.

Conclusion

This work gives explicit expansions for Gauss hypergeometric polynomials parameterized by binomial indices and arbitrary complex values, resulting in nontrivial and widely general combinatorial identities. The methodology, grounded in Faà di Bruno’s formula and specializations of Bell polynomials, synthesizes combinatorial enumeration with analytic hypergeometric function theory. The derived identities not only generalize previous results but provide templates for broad further investigation in both analytic and algebraic combinatorics (2607.10643).