These theories introduce basic concepts and proofs about discrete summation: shifts, formal summation, falling factorials and stirling numbers. As proof of concept, a simple summation conversion is provided. 1 Stirling numbers of first and second kind theory Stirling imports Main begin 1.1 Stirling numbers of the second kind fun Stirling :: nat ⇒ nat ⇒ nat where Stirling 0 0 = 1 | Stirling 0 (S...

Abstract. We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan’s 1ψ1 summation, Bailey’s 2ψ2 transformations, and Bailey’s 6ψ6 summation. Corresponding Author: William Y. C. Chen, Email: [email protected] AMS Cl...