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Zbl 0985.11048
Andrews, George E.
Umbral calculus, Bailey chains, and pentagonal number theorems.
(English)
[J] J. Comb. Theory, Ser. A 91, No.1-2, 464-475 (2000). ISSN 0097-3165

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 $$\multline \sum_{i,j,k\ge 0}{q^{i^2+j^2 +k^2}\over (q;q)_{i+j-u} (q;q)_{i+k-j} (q;q)_{j+k-i}}=\\ ={\sum^\infty_{n,m,p= -\infty}(-1)^{n+m+p} q^{n( 3n-1)/2 +m(3m-1)/2+p(3p-1)/2 +nm+np + mp}\over \prod^\infty_{n=1}(1-q^n)^3}. \endmultline$$ 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.
[Shaun Cooper (Auckland)]
MSC 2000:
*11P81 Elementary theory of partitions
05A40 Umbral calculus
33D80 Connections with groups, algebras and related topics
33D15 Basic hypergeometric functions of one variable

Keywords: umbral calculus; Bailey chain; Rogers-Ramanujan identity; pentagonal number theorem; Jacobi triple product identity; Macdonald identities

Cited in: Zbl 1192.33001 Zbl 1034.33009

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