In Praise of an Elementary Identity of Euler

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu–Vandermonde sum, the Pfaff–Saalschütz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.

Download Full-text

Bisymmetric functions, Macdonald polynomials and basic hypergeometric series

Compositio Mathematica ◽

10.1112/s0010437x07003211 ◽

2008 ◽

Vol 144 (2) ◽

pp. 271-303 ◽

Cited By ~ 14

Author(s):

S. Ole Warnaar

Keyword(s):

Hypergeometric Series ◽

Basic Hypergeometric Series ◽

Macdonald Polynomials ◽

Difference Operators ◽

Jack Polynomials ◽

Selberg Integral ◽

Binomial Theorem ◽

Alternating Sign Matrices ◽

Selberg Integrals ◽

New Type

AbstractA new type of $\mathfrak {sl}_3$ basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key ingredient in the $\mathfrak {sl}_3$ basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the $\mathfrak {sl}_3$ Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for $\mathfrak {sl}_3$ series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new $\mathfrak {sl}_3$ Selberg integral for Jack polynomials.

Download Full-text

The $q$-Binomial Theorem and two Symmetric $q$-Identities

The Electronic Journal of Combinatorics ◽

10.37236/1727 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 2

Author(s):

Victor J. W. Guo

Keyword(s):

Hypergeometric Series ◽

Basic Hypergeometric Series ◽

Binomial Theorem ◽

Cambridge University ◽

Special Cases

We notice two symmetric $q$-identities, which are special cases of the transformations of $_2\phi_1$ series in Gasper and Rahman's book (Basic Hypergeometric Series, Cambridge University Press, 1990, p. 241). In this paper, we give combinatorial proofs of these two identities and the $q$-binomial theorem by using conjugation of $2$-modular diagrams.

Download Full-text