Summation Formulae for Binomial Moments
Marta Na Chen, Wenchang Chu
Abstract
By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined. Several explicit summation formulae are established.
Table of Contents
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Summation Formulae for Binomial Moments
Authors
Marta Na Chen, Wenchang Chu
Abstract
By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined. Several explicit summation formulae are established.
Paper Structure
This paper contains 12 sections, 15 theorems, 55 equations.
Table of Contents
- Introduction and Outline
- Positive Sums A_m(n):=∑_k=1^n2nn-kk^m
- Moments of even powers
- Moments of odd powers
- Alternating Sums B_m(n):=∑_k=1^n(-1)^k-12nn-kk^m
- Moments of even powers
- Moments of odd powers
- Alternating Sums C_m(n):=∑_k=1^n(-1)^k-1[0mm02nn-k]k^m
- Moments of even powers
- Moments of odd powers
- Positive Sums D_m(n):=∑_k=1^n[0mm02nn-k]k^m
- Concluding Comments and Problems
Key Result
Lemma 1
Let $m$ be a nonnegative integer and $\{x,y\}$ two indeterminates. Then the following algebraic equation holds where $\sigma_{m,\ell}(y)$ is a complete symmetric function and admits the explicit expression:
Theorems & Definitions (15)
- Lemma 1: Chen and Chu kn:chen+w
- Theorem 2: $m,n\in\mathbb{N}$
- Corollary 3
- Theorem 4: $m,n\in\mathbb{N}$
- Corollary 5
- Theorem 6: $m,n\in\mathbb{N}$
- Corollary 7
- Theorem 8: $m,n\in\mathbb{N}$
- Corollary 9: $n\in\mathbb{N}$
- Theorem 10: $m,n\in\mathbb{N}$
- ...and 5 more