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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 \[ 1= {\sum^\infty_{n= -\infty} (-1)^nq^{n(3n-1)/2} \over\prod^\infty_{n=1} (1-q^n)}. \] In the paper under review, the author obtains the new pentagonal number theorems: \[ \sum^\infty_{n=1} {q^{2n^2} \over(q;q)_{2n}} ={\sum^\infty_{m,n=-\infty} (-1)^{n+m}q^{n(3n-1)/2 +m(3m-1)/2 +nm}\over \prod^\infty_{n=1} (1-q^n)^2} \] and \[ \begin{split} \sum_{i,j,k\geq 0}{q^{i^2+j^2 +k^2}\over (q;q)_{i+j-u} (q;q)_{i+k-j} (q;q)_{j+k-i}}=\\ ={\sum^\infty_{n,m,p= -\infty}(-1)^{n+m+p} q^{n( 3n-1)/2 +m(3m-1)/2+p(3p-1)/2 +nm+np + mp}\over \prod^\infty_{n=1}(1-q^n)^3}. \end{split} \] 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.

11P81 Elementary theory of partitions
05A40 Umbral calculus
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
Full Text: DOI
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