## 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, $${}_r\phi_s$$
Full Text:

### References:

 [1] Andrews, G.E., () [2] Andrews, G.E., Multiple series rogers – ramanujan type identities, Pac. J. math., 141, 267-283, (1984) · Zbl 0547.10012 [3] Andrews, G.E., q-series, their development and application in analysis, number theory, combinatorics, physics and computer algebra, CBMS regional conference lecture series, 66, (1986), Amer. Math. Soc Providence [4] Berkovich, A.; McCoy, B.M.; Schilling, A., N=2 supersymmetry and bailey pairs, Phys. A, 228, 33-62, (1996) [5] Dickson, L.E., History of the theory of numbers, Carnegie inst. of Washington, 2, (1920) [6] Gasper, G.; Rahman, M., () [7] Humbert, G., Démonstration analytique d’une formule de Liouville, Bull. sci. math. (2), 34, 29-31, (1910) · JFM 41.0510.03 [8] Macdonald, I.G., Affine root systems and Dedekind’s η-function, Invent. math., 15, 91-143, (1972) · Zbl 0244.17005 [9] Paule, P., The concept of bailey chains, Sém. lothar. combin. B, 18f, 24, (1987) · Zbl 0978.05521 [10] Rogers, L.J., On two theorems of combinatory analysis and some allied identities, Proc. London math. soc. (2), 16, 315-336, (1917) · JFM 46.0109.01 [11] Rota, G.C., Ten mathematics problems I will never solve, DMV mitteilungen, 2, 45-52, (1998) · Zbl 1288.00005 [12] Stanton, D., An elementary approach to the Macdonald identities, () · Zbl 0736.05010 [13] Winquist, L., An elementary proof of p(11m+6)≡0 (mod11), J. combin. theory, 6, 56-59, (1969) · Zbl 0241.05006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.