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Finite sums associated with some polynomial identities

Kunle Adegoke, Robert Frontczak, Karol Gryszka

TL;DR

Proposes a general framework to derive finite sums involving binomial coefficients and harmonic numbers from polynomial identities. The approach starts from a polynomial identity equating two finite sums, then multiplies by powers of (1−t) and t, uses the Beta integral, and differentiates with respect to parameters to generate harmonic-number terms. The framework is applied to several prominent identities (Knuth–Boyadzhiev, Frontczak, Dattoli, and Chebyshev), yielding a large set of closed-form sums and corrections, and it suggests a path to many further results. Overall, the method provides a unified toolkit for transforming classical polynomial identities into concrete combinatorial sums with broad applicability.

Abstract

In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and also harmonic numbers and squared harmonic numbers. We apply the framework to discuss combinatorial sums associated with some prominent polynomial identities from the recent past.

Finite sums associated with some polynomial identities

TL;DR

Proposes a general framework to derive finite sums involving binomial coefficients and harmonic numbers from polynomial identities. The approach starts from a polynomial identity equating two finite sums, then multiplies by powers of (1−t) and t, uses the Beta integral, and differentiates with respect to parameters to generate harmonic-number terms. The framework is applied to several prominent identities (Knuth–Boyadzhiev, Frontczak, Dattoli, and Chebyshev), yielding a large set of closed-form sums and corrections, and it suggests a path to many further results. Overall, the method provides a unified toolkit for transforming classical polynomial identities into concrete combinatorial sums with broad applicability.

Abstract

In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and also harmonic numbers and squared harmonic numbers. We apply the framework to discuss combinatorial sums associated with some prominent polynomial identities from the recent past.

Paper Structure

This paper contains 11 sections, 35 theorems, 141 equations.

Table of Contents

  1. Motivation and Preliminaries
  2. Definitions and required identities
  3. General results for arbitrary polynomial identities
  4. A simple application
  5. Applications
  6. Sums associated with polynomial identities of Knuth-Boyadzhiev and Frontczak
  7. Sums derived from polynomial identities involving $H_n^2$ and $H_n^{(2)}$
  8. Sums associated with an identity of Dattoli et al.
  9. Sums associated with an identity for the Chebyshev polynomials of the second kind
  10. Additional results
  11. Conclusion

Key Result

Lemma 1

For $u,v\in\mathbb C\setminus\mathbb Z^{-}$, we have

Theorems & Definitions (66)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Example 4
  • Lemma 5
  • Lemma 6
  • Corollary 7
  • ...and 56 more