On a combinatorial identity of Chaundy and Bullard

2022/05/01 by Alzer, Horst, Kouba, Omran
#05A19 #33B15 #33B20 #39B22 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2205.00480

Abstract

We give two new proofs of the Chaundy-Bullard formula (1-x)ⁿ⁺¹ ∑_k=0^m n+k\choose k x^k +x^m+1∑_k=0ⁿ m+k\choose k (1-x)^k=1 and we prove the "twin formula" \frac (1-x)⁽ⁿ⁺¹⁾(n+1)! ∑_k=0^m (n+1)/(n+k+1) \frac x^(k)k! + \frac x^(m+1)(m+1)! ∑_k=0ⁿ (m+1)/(m+k+1) \frac (1-x)^(k)k!=1, where z⁽ⁿ⁾ denotes the rising factorial. Moreover, we present identities involving the incomplete beta function and a certain combinatorial sum.

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