Related papers: An Alternative Generating Function for $k$-Regular…

The Kostka-Foulkes polynomials $K$ related to a root system $\phi $ can be defined as alternated sums running over the Weyl group associated to $\phi .$ By restricting these sums over the elements of the symmetric group when $% \phi $ is of…

In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of…

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…

In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted $(h,q)$-extension of Euler polynomials and numbers, by using $p$-adic q-deformed fermionic integral on $\Bbb Z_p$. By applying their generating functions,…

We consider $\mathcal{N}=2$ supersymmetric pure gauge theories on toric K\"ahler manifolds, with particular emphasis on $\mathbb{CP}^2$. By choosing a vector generating a $U(1)$ action inside the torus of the manifold, we construct…

In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…

The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to…

We consider generating functionals for computing correlators in quantum field theories with random potentials. Examples of such theories include condensed matter systems with quenched disorder (e.g. spin glass) or cosmological systems in…

We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential…

The Bessel function of the first kind $J_{N}\left(kx\right)$ is expanded in a Fourier-Legendre series, as is the modified Bessel functions of the first kind $I_{N}\left(kx\right)$. The purpose of these expansions in Legendre polynomials was…

Using non-archimedean q-integrals on Zp defined in [15, 16], we define a new Changhee q-Euler polynomials and numbers which are different from those of Kim [7] and Carlitz [2]. We define generating functions of multiple q-Euler numbers and…

We present Euler-type recurrence relations for some partition functions. Some of our results provide new recurrences for the number of unrestricted partitions of $n$, denote by $p(n)$. Others establish recurrences for partition functions…

We present some Euler-type recurrences for the partition function $p(n)$.

Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in…

This paper will primarily present a method of proving generating function identities for partitions from linked partition ideals. The method we introduce is built on a conjecture by George Andrews and that those generating functions satisfy…

We build upon the work by Bessenrodt and Ono, as well as Beckwith and Bessenrodt concerning the combined additive and multiplicative behavior of the $k$-regular partition functions $p_k(n)$. Our focus is on addressing the solutions of the…

We evaluate $q$-Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the $m$-version of the Rogers-Ramanujan identities. We also prove several generating…

We obtain a new family of relations satisfied by the partition function. In contrast with most partition relations, these involve non-trivial roots of unity. We present two proofs, one using the fact that the discriminant modular form is a…

We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers-Ramanujan…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…