TL;DR
This paper presents a concise proof of Knuth's old sum, introduces generalizations, and derives new polynomial and combinatorial identities using binomial theorem and Beta function-based integration.
Contribution
It offers a new, streamlined proof of Knuth's old sum and extends it with generalizations and novel identities not previously documented.
Findings
A short proof of Knuth's old sum is provided.
New polynomial and combinatorial identities are derived.
Generalizations of the original sum are introduced.
Abstract
We give a short proof of the well-known Knuth's old sum and provide some generalizations. Our approach utilizes the binomial theorem and integration formulas derived using the Beta function. Several new polynomial identities and combinatorial identities are derived.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · semigroups and automata theory · Parallel Computing and Optimization Techniques