Inversion techniques and combinatorial identities. Strange evaluations of basic hypergeometric series
Inversion techniques and combinatorial identities. Jackson’s -analogue of the Dougall-Dixon theorem and the dual formulae
Hypergeometric series and the Riemann zeta function
Quintuple products and Ramanujan's partition congruence p(11n+6) ≡ 0(mod 11)
Duplicate inverse series relations and hypergeometric evaluations with
The Gould-Hsu (1973) inverse series relations have been systematically applied to the research of hypergeometric identities. Their duplicate version is established and used to demonstrate several terminating -summation formulas. Further hypergeometric evaluations with the same variable are obtained through recurrence relations.
Divided differences and symmetric functions
L'operatore di differenze multivariate è utilizzato per stabilire varie formule di somme riguardanti le funzioni simmetriche, le quali hanno uno stretto legame con le identità del «termine costante».
Jacobi's Triple Product Identity and the Quintuple Product Identity
The simplest proof of Jacobi's triple product identity originally due to Cauchy (1843) and Gauss (1866) is reviewed. In the same spirit, we prove by means of induction principle and finite difference method, a finite form of the quintuple product identity. Similarly, the induction principle will be used to give a new proof of another algebraic identity due to Guo and Zeng (2005), which can be considered as another finite form of the quintuple product identity.
Reciprocal Formulae on Binomial Convolutions of Hagen-Rothe Type
By means of duplicate inverse series relations, we investigate dual relations of four binomial convolution identities. Four classes of reciprocal formulae on binomial convolutions of Hagen-Rothe type are established. They reflect certain “reciprocity” on the Hagen-Rothe-like convolutions in the sense that each binomial summation involved has no closed form in general, but their sum and difference in pairs do have simple expressions in a single term of binomial coefficients.
Further determinant identities related to classical root systems
By introducing polynomials in matrix entries, six determinants are evaluated which may be considered extensions of Vandermonde-like determinants related to the classical root systems.
Funzione generatrice e polinomi incompleti di Fibonacci e Lucas
I numeri incompleti di Fibonacci e di Lucas, introdotti da Filipponi (1996), sono entrambi generalizzati in forma di polinomi. Le loro funzioni generatrici ridondanti, naturali e condizionate sono stabilite attraverso serie formali di potenze. Le funzioni generatrici relative alle sequenze di numeri dovute a Pinter e Srivastava (1999) sono contenute come casi particolari.
Approach of -Derivative Operators to Terminating -Series Formulae
The -derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.
Identità Binomiali e Numeri Armonici
Numerose identità classiche sui numeri armonici sono mostrate tramite l'operatore di derivazione di Newton ai coefficienti binomiali.
Inverse series relations, formal power series and Blodgett-Gessel's type binomial identities.
Basic hypergeometric identities: An introductory revisiting through the Carlitz inversions.
The Cauchy double alternant and divided differences.
Elementary proofs for convolution identities of Abel and Hagen-Rothe.
A binomial coefficient identity associated with Beukers' conjecture on Apéry numbers.
An algebraic summation over the set of partitions and some strange evaluations
Inverse series relations, formal power series and Blodgett-Gessel's type binomial identities.
A pair of simple bivariate inverse series relations are used by embedding machinery to produce several double summation formulae on shifted factorials (or binomial coefficients), including the evaluation due to Blodgett-Gessel. Their q-analogues are established in the second section. Some generalized convolutions are presented through formal power series manipulation.
Partial sums of two quartic -series.
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