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Umbral calculus, Bailey chains, and pentagonal number theorems. (English) Zbl 0985.11048

Euler’s pentagonal number theorem may be written as

1= n=- (-1) n q n(3n-1)/2 n=1 (1-q n )·

In the paper under review, the author obtains the new pentagonal number theorems:

n=1 q 2n 2 (q;q) 2n = m,n=- (-1) n+m q n(3n-1)/2+m(3m-1)/2+nm n=1 (1-q n ) 2


i,j,k0 q i 2 +j 2 +k 2 (q;q) i+j-u (q;q) i+k-j (q;q) j+k-i == n,m,p=- (-1) n+m+p q n(3n-1)/2+m(3m-1)/2+p(3p-1)/2+nm+np+mp n=1 (1-q n ) 3 ·

The author traces the origin of his ideas back through umbral calculus, Rogers’ second proof of the Rogers-Ramanujan identities and Liouville’s eighteen papers. The paper concludes with some ideas for possible future applications of the methods.

11P81Elementary theory of partitions
05A40Umbral calculus
33D80Connections of basic hypergeometric functions with groups, algebras and related topics
33D15Basic hypergeometric functions of one variable, r φ s