[Paper Review] New identities for a sum of products of the Kummer functions

Sergei Kalmykov, Dmitrii Karp|arXiv (Cornell University)|Oct 22, 2017

Mathematical functions and polynomials15 references3 citations

TL;DR

This paper generalizes a known identity for products of Kummer functions by introducing an additional integer shift in the lower parameter, extending a formula by Feng, Kuznetsov, and Yang. The key result is a new transformation formula for terminating Clausen-type hypergeometric series ${}_{3}F_2(1)$, which enables a linearization identity for differences of Kummer function products, expressed as a polynomial in the argument. The derivation relies on a novel summation formula for non-terminating ${}_{3}F_2(1)$ with specific parameter differences.

ABSTRACT

Recently, Feng, Kuznetsov and Yang discovered a very general reduction formula for a sum of products of the generalized hypergeometric functions (J. Math. Anal. Appl. 443(2016), 116--122). The main goal of this note is to present a generalization of a particular case of their identity when the generalized hypergeometric function is reduced to the Kummer function 1F1. Our generalized formula contains an additional integer shift in the bottom parameter of the Kummer function. The key ingredient of the proof is a summation formula for the Clausen series 3F2(1) with two integral parameter differences of opposite sign. In the ultimate section of the paper we prove another formula for a particular product difference of the Kummer functions in terms of a linear combination of these functions.

Motivation & Objective

• To extend a known reduction formula for products of Kummer functions by introducing an additional integer shift in the lower parameter.
• To generalize the identity of Feng, Kuznetsov, and Yang for ${}_{1}F_1$ functions by incorporating two independent integer shifts.
• To establish a new transformation formula for terminating ${}_{3}F_2(1)$ series with opposite-sign parameter differences.
• To derive a linearization identity for a difference of products of Kummer functions in terms of a polynomial in the argument.
• To provide a new summation formula for non-terminating ${}_{3}F_2(1)$ with one top parameter less than a bottom parameter.

Proposed method

• Derives a new summation formula for non-terminating ${}_{3}F_2(1)$ with one top parameter less than a bottom parameter, using hypergeometric function duality and transformation identities.
• Applies this summation to obtain a transformation formula for terminating ${}_{3}F_2(1)$ series with two integral parameter differences of opposite sign.
• Uses the transformation formula as a key ingredient to prove the generalized identity for products of Kummer functions with two integer shifts in the lower parameter.
• Employs the Kummer transformation $ {}_{1}F_{1}(a;b;x) = e^x {}_{1}F_{1}(b-a;b;-x) $ to re-express the generalized identity in terms of $ t = -x $.
• Applies generating function techniques to derive a linearization identity for a difference of Kummer function products, reducing it to a polynomial expression.
• Validates the results through explicit examples with specific integer shifts $ m_1, m_2 $, yielding closed-form polynomial identities.

Experimental results

Research questions

• RQ1 Can the known identity for products of Kummer functions be generalized by introducing a second integer shift in the lower parameter?
• RQ2 What transformation formula applies to terminating ${}_{3}F_2(1)$ series with two integral parameter differences of opposite sign?
• RQ3 Is there a new summation formula for non-terminating ${}_{3}F_2(1)$ with one top parameter less than a bottom parameter?
• RQ4 Can a linearization identity be derived for a difference of products of Kummer functions, expressing it as a polynomial in the argument?
• RQ5 How do the parameters $ m_1 $ and $ m_2 $ influence the degree and coefficients of the resulting polynomial in the generalized identity?

Key findings

• The generalized identity introduces two independent integer shifts $ m_1, m_2 $ in the lower parameters of the Kummer functions, extending the known formula by Feng, Kuznetsov, and Yang.
• A new transformation formula for terminating ${}_{3}F_2(1)$ series with parameter differences $ m_1 $ and $ -m_2 $ is derived, enabling the main result.
• A novel summation formula for non-terminating ${}_{3}F_2(1)$ is established, valid when one top parameter is less than a bottom parameter, with parameters differing by integers.
• The linearization identity (Theorem 7) expresses a difference of products of Kummer functions as a polynomial in $ t $, with coefficients depending on $ eta, eta+1, eta+2 $, and $ eta+3 $.
• Explicit examples (e.g., $ m_1=1, m_2=0 $) yield identities such as $ \frac{1-eta}{1-eta-1} {}_{1}F_{1}(\beta; \gamma; t) {}_{1}F_{1}(2-\beta; 2-\gamma; -t) - \cdots = 1 $, confirming the polynomial structure.
• For $ m_1=2, m_2=4 $, the resulting polynomial is $ (\gamma-4)_3 + (2\beta - \gamma)(\gamma-3)t $, demonstrating the dependence on parameter shifts and Pochhammer symbols.