The Ramanujan Journal

A result of Stolarsky is extended to show that there are infinitely many irreducible fourth order linear recurrences satisfied by sequences of pairs $$(\left\lfloor {i\Phi } \right\rfloor ,\left\lfloor {i\Phi ^2 } \right\rfloor )$$ , where i is a natural number and φ is the golden ratio $$\frac{1}{2}(1 + \sqrt {5)} $$ . It is also proved that the characteristic polynomial of every such recurrence factorizes non-trivially if φ is adjoined to the rationals.

In this paper, we give a closed-form expression of the inversion and the connection coefficients for general basic hypergeometric polynomial sets using some known inverse relations. We derive expansion formulas corresponding to all the families within the q-Askey scheme and we connect some d-orthogonal basic hypergeometric polynomials.

In this paper, by the Bernoulli numbers and the exponential complete Bell polynomials, we establish two general asymptotic expansions on the hyperfactorial functions $$\prod _{k=1}^nk^{k^q}$$ and the generalized Glaisher–Kinkelin constants $$A_q$$ , where the coefficient sequences in the expansions can be determined by recurrences. Moreover, the explicit expressions of the coefficient sequences are presented and some special asymptotic expansions are discussed. It can be found that some well-known or recently published asymptotic expansions on the factorial function n!, the classical hyperfactorial function $$\prod _{k=1}^nk^k$$ , and the classical Glaisher–Kinkelin constant $$A_1$$ are special cases of our results, so that we give a unified approach to dealing with such asymptotic expansions.

This paper is devoted to the study of sequences in overpartitions and their relation to 2-color partitions. An extensive study of a general class of double series is required to achieve these ends.

In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and $$r_{s, k}(n)$$ , the number of representations of a positive integer n as a sum of $$s\;k$$ -th positive integral powers, based on the recently proved Main Conjecture in Vinogradov’s Mean Value Method.

Let k be an integer with $$k\ge 3$$ and $$\eta $$ be any real number. Suppose that $$\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu $$ are non-zero real numbers, not all of the same sign and $$\lambda _1/\lambda _2$$ is irrational. It is proved that the inequality $$|\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+\eta |<(\max \ p_j)^{-\sigma }$$ has infinitely many solutions in prime variables $$p_1, p_2, \ldots , p_5$$ , where $$0<\sigma <\frac{1}{16}$$ for $$k=3,\ 0<\sigma <\frac{5}{3k2^k}$$ for $$4\le k\le 5$$ and $$0<\sigma <\frac{40}{21k2^k}$$ for $$k\ge 6$$ . This gives an improvement of an earlier result.

Recently, Mc Laughlin proved some results on vanishing coefficients in the series expansions of certain infinite q-products for arithmetic progressions modulo 5, modulo 7 and modulo 11 by grouping the results into several families. In this paper, we prove some new results on vanishing coefficients for arithmetic progressions modulo 7, which are not listed by Mc Laughlin. For example, we prove that if $$t \in \{1,2,3\}$$ and the sequence $$\{A_n\}$$ is defined by $$\sum _{n=0}^{\infty }A_nq^n := (-q^t,-q^{7-t};q^7)_{\infty }(q^{7-2t},q^{7+2t};q^{14})^3_{\infty },$$ then $$A_{7n+4t}=A_{7n+6t^2+4t}=0$$ for all n. Also, we prove four families of results with negative signs for arithmetic progressions modulo 7 classified by Mc Laughlin.