• Ramanujan J. (IF 1.01) Pub Date : 2020-05-29
Jean-Marc Deshouillers, Adrián Ubis

We estimate the maximal number of integral points which can be on a convex arc in $${\mathbb {R}}^2$$ with given length, minimal radius of curvature and initial slope.

更新日期：2020-05-29
• Ramanujan J. (IF 1.01) Pub Date : 2020-05-15
Nancy S. S. Gu, Chen-Yang Su

An overpartition is a partition in which the first occurrence of a number may be overlined. For an overpartition $$\lambda$$, let $$\ell (\lambda )$$ denote the largest part of $$\lambda$$, and let $$n(\lambda )$$ denote its number of parts. Then the $$M_2$$-rank of an overpartition is defined as \begin{aligned} M_2\text {-rank}(\lambda ):=\left\lceil \frac{\ell (\lambda )}{2}\right\rceil -n(\lambda 更新日期：2020-05-15 • Ramanujan J. (IF 1.01) Pub Date : 2020-05-14 L. Deleaval, N. Demni In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as an infinite series of confluent Horn functions. The key ingredient leading to this expression is an extension of an identity involving Gegenbauer polynomials proved in a previous paper by the authors, together with the use of the Poisson kernel for these 更新日期：2020-05-14 • Ramanujan J. (IF 1.01) Pub Date : 2020-05-14 Raimundas Vidunas Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations have already been considered for algebraic $${}_{2}\text{ F }_{1}$$-functions. This article presents Darboux evaluations for the classical case of $${}_{3}\text{ F }_{2}$$-functions with the projective 更新日期：2020-05-14 • Ramanujan J. (IF 1.01) Pub Date : 2020-05-13 Nadir Murru The polynomial Pell equation is\begin{aligned} P^2 - D Q^2 = 1, \end{aligned}$$where D is a given integer polynomial and the solutions P, Q must be integer polynomials. A classical paper of Nathanson (Proc Am Math Soc 86:89–92, 1976) solved it when $$D(x) = x^2 + d$$. We show that the Rédei polynomials can be used in a very simple and direct way for providing these solutions. Moreover, this approach 更新日期：2020-05-13 • Ramanujan J. (IF 1.01) Pub Date : 2020-05-12 Cristina Ballantine, Mircea Merca The main result of this paper is an identity expressing the r-Stirling number of the first kind as a sum involving binomial coefficients and the Möbius function of the set-partition lattice. We provide three different proofs of this result: analytic, inductive, and combinatorial. 更新日期：2020-05-12 • Ramanujan J. (IF 1.01) Pub Date : 2020-05-09 Edward Y. S. Liu, Helen W. J. Zhang Let $$\overline{p}(n)$$ denote the overpartition function. Engel showed that for $$n\ge 2$$, $$\overline{p}(n)$$ satisfy the Turán inequalities, that is, $$\overline{p}(n)^2-\overline{p}(n-1)\overline{p}(n+1)>0$$ for $$n\ge 2$$. In this paper, we prove several inequalities for $$\overline{p}(n)$$. Moreover, motivated by the work of Chen, Jia and Wang, we find that the higher order Turán inequalities 更新日期：2020-05-09 • Ramanujan J. (IF 1.01) Pub Date : 2020-05-09 Wei Cao, Shaofang Hong Let $${\mathbb {F}}_{q}$$ be the finite field of q elements and $$\chi _{1},\ldots ,\chi _{n}$$ the multiplicative characters of $${\mathbb {F}}_{q}$$. Given a Laurent polynomial $$f(X)\in {\mathbb {F}}_q[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$$, the corresponding L-function is defined to be$$\begin{aligned} L^{*}(\chi _{1},\ldots ,\chi _{n},f;T) =\exp \Big (\sum \nolimits _{h=1}^{\infty }S^{*}_{h}(\chi

更新日期：2020-05-09
• Ramanujan J. (IF 1.01) Pub Date : 2020-05-09
Mohamed El Bachraoui

Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.

更新日期：2020-05-09
• Ramanujan J. (IF 1.01) Pub Date : 2019-05-31
Yoshihiro Takeyama

We introduce derivations on the algebra of multiple harmonic q-series and show that they generate linear relations among the q-series which contain the derivation relations for a q-analogue of multiple zeta values due to Bradley. As a byproduct, we obtain Ohno-type relations for finite multiple harmonic q-series at a root of unity.

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-07-03
Shingo Sugiyama, Masao Tsuzuki

In this paper, we give an optimal estimate of an average of Hurwitz class numbers. As an application, we give an equidistribution result of the family $$\Big \{\frac{t}{2q^{\nu /2}} \ | \ \nu \in {{\mathbb {N}}}, t \in {{\mathbb {Z}}}, |t|\leqslant 2q^{\nu /2}\Big \}$$ with q prime, weighted by Hurwitz class numbers. This equidistribution produces many asymptotic relations among Hurwitz class numbers

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-05-29
Sanhua Li, Yingchun Cai

Let $$2< c < \frac{52}{25}$$. In this paper, it is proved that for any sufficiently large real number N, the Diophantine inequality $$|p_1^{c} + p_2^{c} + p_3^{c} + p_4^{c} + p_5^{c} - N|< N^{-\frac{9}{10c}(\frac{52}{25}-c)}$$ is solvable in primes $$p_1,\cdots ,p_5$$. This result constitutes an improvement upon that of Baker and Weingartner.

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-02-27
Victor J. W. Guo

We give q-analogues of three Ramanujan-type series for $$1/\pi$$ from q-analogues of ordinary WZ pairs. The first one is a new q-analogue of the following Ramanujan’s formula for $$1/\pi$$: \begin{aligned} \sum _{n=0}^\infty \frac{6n+1}{256^n}{2n\atopwithdelims ()n}^3=\frac{4}{\pi }, \end{aligned} of which another q-analogue was given by the author and Liu early and reproved by different authors

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-07-11
Chris Jennings-Shaffer, Dillon Reihill

We give asymptotic expansions for the moments of the $$M_2$$-rank generating function and for the $$M_2$$-rank generating function at roots of unity. For this we apply the Hardy–Ramanujan circle method extended to mock modular forms. Our formulas for the $$M_2$$-rank at roots of unity lead to asymptotics for certain combinations of N2(r, m, n) (the number of partitions without repeated odd parts of

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-08-21
Shigeaki Tsuyumine

Let $$f(z)=\sum _{n=0}^{\infty }a_{n}{\mathbf {e}}(nz),g(z)=\sum _{n=0}^{\infty }b_{n}{\mathbf {e}}(nz)\ ({\mathbf {e}}(z)=e^{2\pi \sqrt{-1}z})$$ be holomorphic modular forms for $$\Gamma _{0}(N)$$ of integral weight or half integral weight, where their weights or characters are not necessarily equal to each other. We show that $$L(s;f,g):=\sum _{n=1}^{\infty }a_{n}{\overline{b}}_{n}n^{-s}$$ extends

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-01-17
Miho Aoki, Yasuhiro Kishi

Let p be a prime number with $$p\equiv 5\ (\mathrm{mod}\ {8})$$. We construct a new infinite family of pairs of imaginary cyclic fields of degree $$(p-1)/2$$ with both class numbers divisible by p. Let $$k_0$$ be the unique subfield of $$\mathbb {Q}(\zeta _p)$$ of degree $$(p-1)/4$$ and $$u_p=(t+b\sqrt{p})/2\,(>1)$$ be the fundamental unit of $$k:=\mathbb {Q}(\sqrt{p})$$. We put $$D_{m,n}:={\mathcal 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2019-03-20 Shoyu Nagaoka, Sho Takemori The mod p kernel of the theta operator on Hermitian modular forms is studied in the case that the base field is the Eisenstein field. 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2019-05-22 Yuzhe Bai, Eugene Gorsky, Oscar Kivinen We give an explicit recursive description of the Hilbert series and Gröbner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers–Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen. 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2020-01-22 Benjamin Edun We investigate finite and infinite nested square root formulas convergent to unity. 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2019-05-17 Fethi Bouzeffour, Wissem Jedidi, Mubariz Garayev In this paper, we give an attempt to extend some arithmetic properties such as multiplicativity and convolution products to the setting of operator theory and we provide significant examples which are of interest in number theory. We also give a representation of the Euler differential operator by means of the Euler totient arithmetic function and idempotent elements of some associative unital algebra 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2019-07-26 Rajeev Kohli Ramanujan recorded four reciprocity formulas for the Rogers–Ramanujan continued fractions. Two reciprocity formulas each are also associated with the Ramanujan–Göllnitz–Gordon continued fractions and a level-13 analog of the Rogers–Ramanujan continued fractions. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2019-11-19 Tanay Wakhare, Christophe Vignat We compute multiple zeta values (MZVs) built from the zeros of various entire functions, usually special functions with physical relevance. In the usual case, MZVs and their linear combinations are evaluated using a morphism between symmetric functions and multiple zeta values. We show that this technique can be extended to the zeros of any entire function, and as an illustration, we explicitly compute 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2019-01-17 Pascal Baseilhac, Xavier Martin, Luc Vinet, Alexei Zhedanov The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey–Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey–Wilson algebra generated by twisted primitive elements of \(\mathfrak U_q(sl(2))$$. The little q-Jacobi operator and a tridiagonalization of it are shown to realize the equitable embedding

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-04-11
K. R. Vasuki, Mahadevaswamy

In this paper, we prove six Ramanujan’s modular equations of septic degree by employing Ramanujan’s $$_1\psi _1$$ summation formula and certain theta function identities.

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2020-01-21
Ksenia Fedosova, Anke Pohl

Let $$\Gamma$$ be a geometrically finite Fuchsian group and suppose that $$\chi :\Gamma \rightarrow {{\,\mathrm{GL}\,}}(V)$$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $$\Gamma$$ with twist $$\chi$$ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-02-13
S. L. Hill, M. N. Huxley, M. C. Lettington, K. M. Schmidt

The jth divisor function $$d_j$$, which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor function$$c_j$$, which counts the ordered factorisations of a positive integer into j factors each of which is greater than or equal to 2, is rather less well

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-05-29
Sonja Žunar

We prove a strengthening of Muić’s integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups and use it to prove a result on non-vanishing of L-functions associated to cusp forms of half-integral weight.

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-08-23
Stella Brassesco, Arnaud Meyroneinc

We consider p(n) the number of partitions of a natural number n, starting from an expression derived by Báez-Duarte in (Adv Math 125(1):114–120, 1997) by relating its generating function f(t) with the characteristic functions of a family of sums of independent random variables indexed by t. The asymptotic formula for p(n) follows then from a local central limit theorem as $$t\uparrow 1$$ suitably with

更新日期：2020-04-18
• Ramanujan J. (IF 1.01) Pub Date : 2019-06-13
Jangwon Ju

For an integer x, an integer of the form $$P_5(x)=\frac{3x^2-x}{2}$$ is called a generalized pentagonal number. For positive integers $$\alpha _1,\dots ,\alpha _k$$, a sum $$\Phi _{\alpha _1,\dots ,\alpha _k}(x_1,x_2,\dots ,x_k)=\alpha _1P_5(x_1)+\alpha _2P_5(x_2)+\cdots +\alpha _kP_5(x_k)$$ of generalized pentagonal numbers is called universal if $$\Phi _{\alpha _1,\dots ,\alpha _k}(x_1,x_2,\dots 更新日期：2020-04-18 • Ramanujan J. (IF 1.01) Pub Date : 2020-01-09 Maxwell Schneider, Robert Schneider We connect a primitive operation from arithmetic—summing the digits of a base-B integer—to q-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of “B-ary” Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series. 更新日期：2020-01-09 • Ramanujan J. Pub Date : null L D Abreu,R Álvarez-Nodarse,J L Cardoso We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier-Bessel series. 更新日期：2019-11-01 • Ramanujan J. Pub Date : null Gaurav Bhatnagar We apply Heine's method-the key idea Heine used in 1846 to derive his famous transformation formula for 2 ϕ 1 series-to multiple basic series over the root system of type A. In the classical case, this leads to a bibasic extension of Heine's formula, which was implicit in a paper of Andrews which he wrote in 1966. As special cases, we recover extensions of many of Ramanujan's 2 ϕ 1 transformations 更新日期：2019-11-01 • Ramanujan J. Pub Date : null Christian Elsholtz,Florian Luca,Stefan Planitzer Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form n = p + 2 2 k + m ! and n = p + 2 2 k + 2 q where m , k ∈ N and p, q are primes. In the opposite direction, Erdős constructed a full arithmetic progression of odd integers none of which is the sum of a prime and a power 更新日期：2019-11-01 • Ramanujan J. Pub Date : null Andrew Granville,Dimitris Koukoulopoulos The Landau-Selberg-Delange method gives an asymptotic formula for the partial sums of a multiplicative function f whose prime values are α on average. In the literature, the average is usually taken to be α with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak 更新日期：2019-11-01 • Ramanujan J. (IF 1.01) Pub Date : 2019-09-23 Julia Q. D. Du, Edward Y. S. Liu, Jack C. D. Zhao Let \(p_k(n)$$ be given by the series expansion of the k-th power of the Euler Product $$\prod _{n=1}^{\infty }(1-q^n)^k=\sum _{n=0}^{\infty }p_k(n)q^{n}$$. By investigating the properties of the modular equations of the second and the third order under the Atkin U-operator, we determine the generating functions of $$p_{8k}(2^{2\alpha } n +\frac{k(2^{2\alpha }-1)}{3})$$$$(1\le k\le 3)$$ and $$p_{3k}(3^{2\beta 更新日期：2019-09-23 • Ramanujan J. (IF 1.01) Pub Date : 2019-09-21 Atul Dixit, Bibekananda Maji A generalization of a beautiful q-series identity found in the unorganized portion of Ramanujan’s second and third notebooks is obtained. As a consequence, we derive a three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews et al., which itself generalizes the famous result of Fokkink 更新日期：2019-09-21 • Ramanujan J. (IF 1.01) Pub Date : 2019-09-05 Patrice Philippon, Biswajyoti Saha, Ekata Saha In this article we study an abelian analogue of Schanuel’s conjecture. This conjecture falls in the realm of the generalised period conjecture of André. As shown by Bertolin, the generalised period conjecture includes Schanuel’s conjecture as a special case. Extending methods of Bertolin, it can be shown that the abelian analogue of Schanuel’s conjecture we consider also follows from André’s conjecture 更新日期：2019-09-05 • Ramanujan J. (IF 1.01) Pub Date : 2019-09-05 Myung-Hwan Kim, Byeong-Kweon Oh, Dayoon Park In this article, we determine all positive definite integral binary quadratic forms that are represented by the sum of k integer squares in an essentially unique way for each integer \(k\ge 4$$.

更新日期：2019-09-05
• Ramanujan J. (IF 1.01) Pub Date : 2019-06-03
Chiranjit Ray, Rupam Barman

Let b(n) denote the number of cubic partition pairs of n. We affirm a conjecture of Lin by proving that \begin{aligned} b(49n+37)\equiv 0 \pmod {49} \end{aligned} for all $$n\ge 0$$. We also prove two congruences modulo 256 satisfied by $$\overline{b}(n)$$, the number of overcubic partition pairs of n. Let $$\overline{a}(n)$$ denote the number of overcubic partition of n. For any positive integer

更新日期：2019-06-03
• Ramanujan J. (IF 1.01) Pub Date : 2019-05-20
Renrong Mao

We prove inequalities between the first moment of the $$M_2$$-rank of overpartitions and the residual crank of such partitions. In order to obtain such inequalities, we need a result on the positivity of certain spt-cranks of overpartitions which were first introduced by Jennings-Shaffer.

更新日期：2019-05-20
• Ramanujan J. (IF 1.01) Pub Date : 2019-05-17
Cristina Ballantine, Mircea Merca

The least r-gap, $$g_r(\lambda )$$, of a partition $$\lambda$$ is the smallest positive integer that does not appear at least r times as a part of $$\lambda$$. In this article, we introduce two new partition functions involving least r-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler’s partition function p(n), polygonal numbers, and the new partition

更新日期：2019-05-17
• Ramanujan J. (IF 1.01) Pub Date : 2019-03-21
Nayandeep Deka Baruah, Mandeep Kaur

Recently, Lin introduced two new partition functions $$\hbox {PD}_{\mathrm{t}}(n)$$ and $$\hbox {PDO}_{\mathrm{t}}(n)$$, which count the total number of tagged parts over all partitions of n with designated summands and the total number of tagged parts over all partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $$\hbox {PD}_{\mathrm{t}}(n)$$

更新日期：2019-03-21
• Ramanujan J. (IF 1.01) Pub Date : 2019-03-20
Jesús Guillera

In a famous paper of 1914 Ramanujan gave a list of 17 extraordinary formulas for the number $$1/\pi$$. In this paper we explain a general method to prove them, based on some ideas of James Wan and some of our own ideas.

更新日期：2019-03-20
• Ramanujan J. (IF 1.01) Pub Date : 2019-03-16
Pascal Baseilhac, Luc Vinet, Alexei Zhedanov

The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second-order q-difference operator that maps polynomials of degree n to polynomials of degree $$n+1$$. It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes

更新日期：2019-03-16
• Ramanujan J. (IF 1.01) Pub Date : 2019-03-15
Dan Romik

We study the Taylor expansion around the point $$x=1$$ of a classical modular form, the Jacobi theta constant $$\theta _3$$. This leads naturally to a new sequence $$(d(n))_{n=0}^\infty =1,1,-1,51,849,-26199,\ldots$$ of integers, which arise as the Taylor coefficients in the expansion of a related “centered” version of $$\theta _3$$. We prove several results about the numbers d(n) and conjecture that

更新日期：2019-03-15
• Ramanujan J. (IF 1.01) Pub Date : 2019-02-26
Tim Huber, Daniel Schultz, Dongxi Ye

Two level 17 modular functions \begin{aligned} r = q^2 \prod _{n=1}^{\infty } (1-q^{n})^{\left( \frac{n}{17} \right) },\qquad s = q^{2} \prod _{n=1}^{\infty } \frac{(1 - q^{17n})^{3}}{(1-q^{n})^{3}} \end{aligned} are used to construct a new class of Ramanujan–Sato series for $$1/\pi$$. The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan’s

更新日期：2019-02-26
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