Double sums associated with binomial transforms
Kunle Adegoke, Robert Frontczak, Karol Gryszka
TL;DR
The paper develops a comprehensive framework for double sums with incomplete binomial inner sums by leveraging binomial transform pairs and a linear transformation operator. Central results express and manipulate such sums through generating-function identities, yielding a wide array of new closed-form identities across Bernoulli, Fibonacci, harmonic, Catalan, and Stirling-number sequences, as well as Horadam-based polynomial families. The authors provide concrete corollaries and numerous specialized identities, including hyperbolic-function forms and relations for classical polynomials, thereby unifying and expanding known transform-based identities. These contributions enhance the toolkit for manipulating binomial-transform-linked sums with broad implications in number theory and combinatorics.
Abstract
In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new identities for double sums involving special numbers like Bernoulli numbers, Fibonacci numbers, harmonic numbers, Catalan numbers and Stirling numbers of the second kind. We also consider families of polynomials like Fibonacci polynomials, Chebyshev polynomials, Bernoulli polynomials, and others. Finally, we state new double sums involving hyperbolic functions.