Applications of an identity of Batır

TL;DR

The paper uses the identity $\sum_{k=1}^n a_k \sum_{j=1}^k b_j = \sum_{p=0}^{n-1}\sum_{k=1}^{n-p} a_{p+k} b_k$ and a variation $\sum_{k=1}^n \sum_{j=0}^{k-1} a_{n-j} b_{k-j} = \sum_{k=1}^n a_k \sum_{j=1}^k b_j$ to derive extensive double-sum identities across number sequences. It develops general results for sums involving $H_n^{(s)}$, $O_n^{(s)}$, $B_n$, $F_n$, $L_n$, and extends to Stirling numbers, Catalan numbers, and gibonacci numbers, including double-sum definitions of Bernoulli numbers and parameterized forms with $r$ and $x$. The work also provides numerous miscellaneous identities with binomial coefficients and transforms, along with applications to binomial transform pairs that connect classical sequences such as Fibonacci, Lucas, and Bernoulli numbers. Overall, the paper expands the toolkit for evaluating and manipulating double sums in combinatorial-number-theoretic contexts and offers a rich set of identities with potential applications in analytic number theory and combinatorics.

Abstract

Based on an interesting identity of Batır we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.

Theorems & Definitions (88)

• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof