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

Andrews, George E.

J. Comb. Theory, Ser. A 91, No. 1-2, 464-475 (2000).

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.

Reviewer: Shaun Cooper (Auckland)

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, \({}_r\phi_s\)

Keywords:

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

References:

[1]	Andrews, G. E., (Rota, G.-C., The Theory of Partitions. The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, 2 (1976), Addison-Wesley: Addison-Wesley Reading) · Zbl 0371.10001
[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. \(q\)-Series, Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, CBMS Regional Conference Lecture Series, 66 (1986), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0594.33001
[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. History of the Theory of Numbers, Carnegie Inst. of Washington, 2 (1920) · JFM 48.0137.02
[6]	Gasper, G.; Rahman, M., (Rota, G.-C., Basic Hypergeometric Series. Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, 35 (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0695.33001
[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, (Stanton, D., \(q\)-Series and Partitions. \(q\)-Series and Partitions, IMA Volumes in Mathematics and Its Applications, 18 (1989), Springer-Verlag: Springer-Verlag Berlin) · 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