Related papers: Euler-like recurrences for smallest parts function…

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…

Building on work of Hardy and Ramanujan, Rademacher proved a well-known formula for the values of the ordinary partition function $p(n)$. More recently, Bruinier and Ono obtained an algebraic formula for these values. Here we study the…

We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a…

We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing…

We construct new smallest parts partition functions and smallest parts crank functions by considering variations of Bailey's Lemma and conjugate Bailey pairs. The functions we introduce satisfy simple linear congruences modulo $3$ and $5$.…

In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…

By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation…

Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight $3/2$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. These coefficients are related to the smallest…

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function $Li_(z)$. The…

Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove…

Recently, Andrews and El Bachraoui considered the number of integer partitions whose smallest part is repeated exactly $k$ times and the remaining parts are not repeated. They presented several interesting results and posed questions…

In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have…

We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain…

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…

Recently, Mertens, Ono, and the third author studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, and have shadows given by classical Shimura theta functions. Here, we construct a…

We prove that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a…

We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions,…

We prove some new cases of local--global compatibility for the Galois representations associated to Hilbert modular forms of low weight (that is, partial weight one).

We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating…

There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular…