Related papers: On partition polynomials and partition functions
We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
Partition functions of certain classes of "spin glass" models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants,…
A classical method for partition generating functions is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n copies of n are presented.
In this paper special values of Bell polynomials are given by using the power series solution of the equation $y^{(k)}=e^{ay}$. In addition, complete and partial exponential autonomous functions, exponential autonomous polynomials,…
The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general…
In this paper, we investigate decompositions of the partition function $p(n)$ from the additive theory of partitions considering the famous M\"{o}bius function $\mu(n)$ from multiplicative number theory. Some combinatorial interpretations…
In the paper, by induction, the Fa\`a di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the…
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem,…
A conformal partition function ${\cal P}_n^m(s)$, which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with {\it self-dual symmetric polynomials} -- reciprocal ${\sf R}^{\{m\}}_ {S_n}$…
We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack…
We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $\beta$. We prove that if $\beta$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $\beta$…
In this paper, we introduce the quaternionic slice polyanalytic functions and we prove some of their properties. Then, we apply the obtained results to begin the study of the quaternionic Fock and Bergman spaces in this new setting. In…
Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. We consider the probabilistic bivariate Bell polynomials associated with Y and the probabilistic bivariate r-Bell polynomials associated…
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
We obtain formulas for the coefficients of positive and negative powers of a partial theta function.
The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These…
In this paper, we obtain upper and lower bounds for the partition function $p(n)$ by using an elementary geometric inequality in Euclidean space, and we extend the method to generalizations of the partition function.
We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and find many theorems of classical number theory arise as particular cases of extremely general combinatorial structure…
We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…