Related papers: Square Series Generating Function Transformations
A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side…
We derive a generalized Rogers generating function and corresponding definite integral, for the continuous $q$-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the…
In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some…
This article deals with three types of mutually inverse series relating Ferrers and associated Legendre functions of arbitrary complex indexes and orders established on the base of integral representations by using a number of generating…
We study two types of series over a real alternative $^*$-algebra $A$. The first type are series of the form $\sum_{n} (x-y)^{\punto n}a_n$, where $a_n$ and $y$ belong to $A$ and $(x-y)^{\punto n}$ denotes the $n$--th power of $x-y$ w.r.t.\…
In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and…
The aim of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function,…
In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these…
Stirling numbers of both kinds are linked to each other via two combinatorial identities due to Schl\"afli and Gould. Using q-analogs of Stirling numbers defined as inversion generating functions, we provide q-analogs of the two identities.…
We compute the generating function for the characters of the irreducible representations of SU(n) whose associated Young diagrams have only two rows with the same number of boxes. The result is a rational determinantal expression in which…
In the paper, with the aid of the Fa\`a di Bruno formula, in terms of central factorial numbers of the second kind, and with the terminology of the Stirling numbers of the second kind, the authors derive several series expansions for any…
We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions…
We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these $q$-double series and deduce that their coefficients are non-negative. This leads to inequalities…
The quantum integer [n]_q is the polynomial 1 + q + q^2 + ... + q^{n-1}, and the sequence of polynomials { [n]_q }_{n=1}^{\infty} is a solution of the functional equation f_{mn}(q) = f_m(q)f_n(q^m). In this paper, semidirect products of…
In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are…
In this paper, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of…
The multiple gamma function $\Gamma_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of…
We reconstruct the quantum enveloping superalgebra ${\bf U}(\mathfrak{gl}_{m|n})$ over $\mathbb Q(v)$ via (finite dimensional) quantum Schur superalgebras. In particular, we obtain a new basis containing the standard generators of ${\bf…
We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we…
This is a note for constructing fundamental invariants and computing the Hilbert series of the invariant subalgebras of tensor products of polynomial rings under the action by a direct product of symmetric groups. Our computation relies on…