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Abstract: Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler''s pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas’ formulas. [Copyright &y& Elsevier]

Abstract: New subsets of symmetric balanced and symmetric correlation immune functions are identified. The method involves interesting relations on binomial coefficients and highlights the combinatorial richness of these classes. As a consequence of our constructive techniques, we improve upon the existing lower bounds on the cardinality of the above sets. We consider higher order correlation immune functions and show how to construct n-variable, third order correlation immune function for each perfect square . [Copyright &y& Elsevier]

Abstract: As a generalization of Calkin''s identity and its alternating form, we compute a kind of binomial identity involving some real number sequences and a partial sum of the binomial coefficients, from which many interesting identities follow. [Copyright &y& Elsevier]

Abstract: By a very simple argument, we prove that if thenOn the basis of this identity, for we construct explicit and such that for any prime we havewhere denotes the Catalan number . For example, when is a prime, we haveandThis paper also contains some new recurrence relations for Catalan numbers. [Copyright &y& Elsevier]

Abstract: We present some simple observations on factors of the q-binomial coefficients, the q-Catalan numbers, and the q-multinomial coefficients. Writing the Gaussian coefficient with numerator n and denominator k in a form such that by the symmetry in k, we show that this coefficient has at least k factors. Some divisibility results of Andrews, Brunetti and Del Lungo are also discussed. [Copyright &y& Elsevier]

Abstract: We show that the universal continued fraction of the Stieltjes-Jacobi type is equivalent to the characteristic series of labelled paths in the plane. The equivalence holds in the set of series in non-commutative indeterminates. Using it, we derive direct combinatorial proofs of continued fraction expansions for series involving known combinatorial quantities: the Catalan numbers, the Bell and Stirling numbers, the tangent and secant numbers, the Euler and Eulerian numbers We also show combinatorial interpretations for the coefficients of the elliptic functions, the coefficients of inverses of the Tchebycheff, Charlier, Hermite, Laguerre and Meixner polynomials. Other applications include cycles of binomial coefficients and inversion formulae. Most of the proofs follow from direct geometrical correspondences between objects. [Copyright &y& Elsevier]

Abstract: We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities—for example, with all parameters equal to 1—have occurred in the literature. The special case with only one parameter reduces to an identity for the divisor generating function, which has received some attention in connection with problems in sorting theory. The general case can be viewed as a duality result, reminiscent of the duality relation for the ordinary multiple zeta function. [Copyright &y& Elsevier]

We provide a probabilistic approach using renewal theory to derive some novel identities involving Eulerian numbers and uniform B-splines. The renewal perspective leads to a unified treatment for the normalized binomial coefficients and the normalized Eulerian numbers when studying their limits of sums, as well as their associated distributions — the binomial distributions (Bernstein polynomials) and the Irwin–Hall distributions (uniform B-splines). We further extend the probabilistic perspective to h -Bernstein polynomials (Pólya–Eggenberger distributions) through conditional renewal processes, and derive new limits of various ways of summing for the two special numbers and associated distributions. The proposed probabilistic unification dramatically simplifies the proofs of some identities, which are far from obvious (such as for h -Bernstein polynomials) or do not otherwise even appear promising (such as for Eulerian numbers). [ABSTRACT FROM AUTHOR]

Abstract: We give a simple proof of Jackson''s terminating q-analogue of Dixon''s identity. [Copyright &y& Elsevier]