Hybrid Proofs of the q-Binomial Theorem and Other Identities

We give “hybrid” proofs of the q-binomial theorem and other identities. The proofs are “hybrid” in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.

[1]  H. Srivastava.  A note on a generalization of a $q$-series transformation of Ramanujan , 1987 .

[2]  S. Ole Warnaar.  Extensions of the well-poised and elliptic well-poised Bailey lemma , 2003 .

[3]  George E. Andrews,et al.  Ramanujan's Lost Notebook: Part I , 2005 .

[4]  Emil Grosswald,et al.  The Theory of Partitions , 1984 .

[5]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[6]  V. Ramamani,et al.  On a partition theorem of Sylvester. , 1972 .

[7]  Mizan Rahman,et al.  An Indefinite Bibasic Summation Formula and Some Quadratic, Cubic and Quartic Summation and Transformation Formulas , 1990, Canadian Journal of Mathematics.

[8]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[9]  Bruce C. Berndt,et al.  Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions , 2010, J. Comb. Theory, Ser. A.

[10]  Andrew V. Sills,et al.  Finite Rogers-Ramanujan Type Identities , 2003, Electron. J. Comb..

[11]  Tom H. Koornwinder,et al.  On Zeilberger's algorithm and its q-analogue: a rigorous description , 1993 .

[12]  Another proof of Bailey’s 6ψ6 summation , 2003, math/0312236.

[13]  Lucy Joan Slater,et al.  Further Identities of the Rogers‐Ramanujan Type , 1952 .

[14]  Arnold Knopfmacher,et al.  An Infinite Family of Engel Expansions of Rogers-Ramanujan Type , 2000, Adv. Appl. Math..

[15]  Ira M. Gessel,et al.  Applications of q-lagrange inversion to basic hypergeometric series , 1983 .

[16]  L. J. Rogers.  Second Memoir on the Expansion of certain Infinite Products , 1893 .

[17]  Arnold Knopfmacher,et al.  Engel Expansions and the Rogers–Ramanujan Identities☆ , 2000 .

[18]  Igor Pak,et al.  Partition bijections, a survey , 2006 .

[19]  George E. Andrews.  $q$-identities of Auluck, Carlitz, and Rogers , 1966 .

[20]  Ramanujan's lost notebook , 2012 .

[21]  Qing-Hu Hou,et al.  Non-Terminating Basic Hypergeometric Series and the q-Zeilberger Algorithm , 2005, Proceedings of the Edinburgh Mathematical Society.

[22]  L. Carlitz,et al.  Some inverse relations , 1973 .

[23]  L. J. Rogers.  On Two Theorems of Combinatory Analysis and Some Allied Identities , 1917 .

[24]  Nancy S. S. Gu,et al.  One-parameter generalizations of Rogers-Ramanujan type identities , 2009, Adv. Appl. Math..

[25]  George E. Andrews.  Identities in combinatorics. II. A $q$-analog of the Lagrange inversion theorem , 1975 .

[26]  Wenchang Chu,et al.  Abel's lemma on summation by parts and basic hypergeometric series , 2007, Adv. Appl. Math..

[27]  George E. Andrews,et al.  q-series : their development and application in analysis, number theory, combinatorics, physics, and computer algebra , 1986 .

[28]  G. Andrews,et al.  The WP‐Bailey Tree and its Implications , 2001, math/0109141.

[29]  Christian Krattenthaler,et al.  A new matrix inverse , 1996 .