q-Binomial Convolution and Transformations of q-Appell Polynomials

In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

Some result for binomial convolution sums of restricted divisor functions

Besge presented the result about the convolution sum of divisor functions. Since then Liouville obtained the generalized version of Besge's formula, which is the binomial convolution sum of divisor functions. In 2004, Hahn obtained the results about the convolution sums of ?d|n(-1)d-1d and ?d|n (-1)n=d-1d. In this paper, we present the results for the binomial con- voltion sums, generalized convolution sums of Hahn, of these divisor functions.

Positivity and continued fractions from the binomial transformation

Given a sequence of polynomials$\{x_k(q)\}_{k \ges 0}$, define the transformation$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$for$n\ges 0$. In this paper, we obtain the relation between the Jacobi continued fraction of the ordinary generating function ofyn(q) and that ofxn(q). We also prove that the transformation preservesq-TPr+1(q-TP) property of the Hankel matrix$[x_{i+j}(q)]_{i,j \ges 0}$, in particular forr= 2,3, implying ther-q-log-convexity of the sequence$\{y_n(q)\}_{n\ges 0}$. As applications, we can give the continued fraction expressions of Eulerian polynomials of typesAandB, derangement polynomials typesAandB, general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. In addition, we also prove the strongq-log-convexity of derangement polynomials typeB, Dowling polynomials and Tanny-geometric polynomials and 3-q-log-convexity of general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. We also present a new proof of the result of Pólya and Szegö about the binomial convolution preserving the Stieltjes moment property and a new proof of the result of Zhu and Sun on the binomial transformation preserving strongq-log-convexity.

Evaluating binomial convolution sums of divisor functions in terms of Euler and Bernoulli polynomials

In this paper, we provide two identities about binomial convolution sums of [Formula: see text] with [Formula: see text], which are expressed in terms of Euler and Bernoulli polynomials. A recent result of Kim, Bayad and Park turns out to be a special case of one of the two identities when [Formula: see text].

Generalized Binomial Convolution of the mth Powers of the Consecutive Integers with the General Fibonacci Sequence

In this paper, we consider Gauthier’s generalized convolution and then define its binomial analogue as well as alternating binomial analogue. We formulate these convolutions and give some applications of them.

Preserving Log-Convexity for Generalized Pascal Triangles

We establish the preserving log-convexity property for the generalized Pascal triangles. It is an extension of a result of H. Davenport and G. Pólya "On the product of two power series", who proved that the binomial convolution of two log-convex sequences is log-convex.