## 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.

### MSC:

 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, $${}_r\phi_s$$
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