• Ramanujan J. (IF 0.79) Pub Date : 2020-11-04
Pramath Anamby, Soumya Das, Ritwik Pal

We prove a result on the distribution of Hecke eigenvalues, $$\mu _F(p^r)$$ (for $$r=1,2$$ or 3) of a non-Saito–Kurokawa lift F of degree 2. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an F, which is the best possible in terms of orders of magnitude.

更新日期：2020-11-04
• Ramanujan J. (IF 0.79) Pub Date : 2020-11-04
S. Capparelli, A. Del Fra, P. Mercuri, A. Vietri

In the work of Alladi et al. (J Algebra 174:636–658, 1995) the authors provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a

更新日期：2020-11-04
• Ramanujan J. (IF 0.79) Pub Date : 2020-10-28
Gonzalo Cao-Labora, Juanjo Rué, Christoph Spiegel

Let $$k\ge 2$$ be a positive integer. We study concentration results for the ordered representation functions $$r^{{ \le }}_k({\mathcal {A}},n) = \# \big \{ (a_1 \le \dots \le a_k) \in {\mathcal {A}}^k : a_1+\dots +a_k = n \big \}$$ and $$r^{{<}}_k({\mathcal {A}},n) = \# \big \{ (a_1< \dots < a_k) \in {\mathcal {A}}^k : a_1+\dots +a_k = n \big \}$$ for any infinite set of non-negative integers $${\mathcal 更新日期：2020-10-30 • Ramanujan J. (IF 0.79) Pub Date : 2020-10-27 Daejun Kim, Byeong-Kweon Oh Let m, n be positive integers with \(m\le n$$. Let $$\kappa (m,n)$$ be the largest integer k such that for any (positive definite and integral) quadratic forms $$f_1,\ldots ,f_k$$ of rank m, there exists a quadratic form of rank n that represents $$f_i$$ for any i with $$1\le i \le k$$. In this article, we determine the number $$\kappa (m,n)$$ for any integer m with $$1\le m\le 8$$, except for the

更新日期：2020-10-30
• Ramanujan J. (IF 0.79) Pub Date : 2020-10-20
Lulu Fang, Jihua Ma, Kunkun Song

Let $$[a_1(x),a_2(x), a_3(x),\ldots ]$$ denote the continued fraction expansion of a real number $$x \in [0,1)$$. This paper is concerned with certain exceptional sets of the Borel–Bernstein Theorem on the growth rate of $$\{a_n(x)\}_{n\geqslant 1}$$. As a main result, the Hausdorff dimension of the set \begin{aligned} E_{\sup }(\psi )=\left\{ x\in [0,1):\ \limsup \limits _{n\rightarrow \infty }\frac{\log 更新日期：2020-10-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-10-20 Hamed Mousavi, Maxie D. Schmidt We build on and generalize recent work on so-termed factorization theorems for Lambert series generating functions. These factorization theorems allow us to express formal generating functions for special sums as invertible matrix transformations involving partition functions. In the Lambert series case, the generating functions at hand enumerate the divisor sum coefficients of $$q^n$$ as $$\sum _{d|n} 更新日期：2020-10-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-10-20 David Klein, Jennifer Kupka We present completions of mock theta functions to harmonic weak Maass forms of weight \(\nicefrac {1}{2}$$ and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $$\nicefrac {1}{2}$$ that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic 更新日期：2020-10-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-10-19 Mircea Merca In this paper we show that the reciprocal of the finite product $$(q;q)_n$$ can be expressed as a sum over all the partitions of n in two different ways. We derive similar results for the reciprocal of $$1-q^n$$. Product identities involving sums over partitions are derived as consequences of these results. 更新日期：2020-10-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-09-27 Yuya Murakami In this paper, we study a continuity of the “values” of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer structure of the continuity of these values with respect to continued fraction expansions and it turns out that it is different from the continuity with respect to Euclidean 更新日期：2020-09-28 • Ramanujan J. (IF 0.79) Pub Date : 2020-09-22 Jin Wang In this paper, by the technique of matrix inversions, we establish a general q-expansion formula of arbitrary formal power series F(z) with respect to the base\begin{aligned} \left\{ z^n\frac{(az)_{n}}{(bz)_{n}}\bigg |n=0,1,2,\ldots \right\} . \end{aligned}$$Some concrete expansion formulas and their applications to q-series identities are presented, including Carlitz’s q-expansion formula and 更新日期：2020-09-23 • Ramanujan J. (IF 0.79) Pub Date : 2020-09-19 Chiranjit Ray, Kalyan Chakraborty Let $${\overline{A}}_{\ell }(n)$$ be the number of overpartitions of n into parts not divisible by $$\ell$$. In this paper, we prove that $${\overline{A}}_{\ell }(n)$$ is almost always divisible by $$p_i^j$$ if $$p_i^{2a_i}\ge \ell$$, where j is a fixed positive integer and $$\ell =p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}$$ with primes $$p_i>3$$. We obtain a Ramanujan-type congruence for $${\overline{A}}_{7}$$ 更新日期：2020-09-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-09-18 Martin Bridgeman We describe a new connection between the dilogarithm function and the solutions of Pell’s equation $$x^2-ny^2 = \pm 1$$. For each solution x, y to Pell’s equation, we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $$x+y\sqrt{n} \in {\mathbb {Z}}[\sqrt{n}]$$. We further show that Ramanujan’s dilogarithm value-identities correspond to an 更新日期：2020-09-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-09-18 Anthony Sofo In this paper we study the representation of integrals whose integrand involves the product of a polylogarithm and an inverse or inverse hyperbolic trigonometric function. We further demonstrate many connections between these integrals and Euler sums. We develop recurrence relations and give some examples of these integrals in terms of Riemann zeta values, Dirichlet values and other special functions 更新日期：2020-09-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-09-18 Huaning Liu, Mengyao Jing Dartyge and Sárközy (partly with other coauthors) introduced pseudorandom measures of subsets, and studied the pseudorandomness of certain subsets related to primitive roots. In this paper we answer a conjecture of Dartyge, Sárközy and Szalay, and study the pseudorandom properties of some subsets constructed by using primitive roots. 更新日期：2020-09-20 • Ramanujan J. (IF 0.79) Pub Date : 2020-09-18 Bernard L. S. Lin, Lin Peng, Pee Choon Toh Stanley introduced a partition statistic $$srank (\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$$, where $$\mathcal {O}(\pi )$$ denote the number of odd parts of the partition $$\pi$$, and $$\pi '$$ is the conjugate of $$\pi$$. Let $$p_i(n)$$ denote the number of partitions of n with srank $$\equiv i\pmod 4$$. Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5:$$\begin{aligned}

更新日期：2020-09-20
• Ramanujan J. (IF 0.79) Pub Date : 2020-09-09
Semyon Yakubovich

Discrete analogs of the classical Kontorovich–Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function $$K_{in}(x), x >0, n \in {\mathbb {N}}, i$$ is the imaginary unit, and incomplete Bessel functions. Several expansions of suitable functions and sequences in terms of these series and integrals are established. As an application

更新日期：2020-09-10
• Ramanujan J. (IF 0.79) Pub Date : 2020-09-09
Dandan Chen, Rong Chen

We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $$\wp (z|\tau )$$ as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular

更新日期：2020-09-10
• Ramanujan J. (IF 0.79) Pub Date : 2020-09-08
Bernard L. S. Lin, Saisai Zheng

Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions

更新日期：2020-09-08
• Ramanujan J. (IF 0.79) Pub Date : 2020-09-08
Mourad E. H. Ismail, Plamen Simeonov

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Itô polynomials. We show how these

更新日期：2020-09-08
• Ramanujan J. (IF 0.79) Pub Date : 2020-08-29
Daeyeol Jeon, Soon-Yi Kang, Chang Heon Kim

We construct bases of the space of harmonic weak Maass forms of weight $$\kappa \in \frac{1}{2}{\mathbb {Z}}$$. Using these bases, we obtain a Shintani lift from a positive integral weight harmonic weak Maass form to a half-integral weight harmonic weak Maass form, which reduces to the classical Shintani lift on the space of cusp forms.

更新日期：2020-08-29
• Ramanujan J. (IF 0.79) Pub Date : 2020-08-28
Andrés Chirre, Oswaldo Velásquez Castañón

In this paper we review the study of the distribution of the zeros of certain approximations for the Ramanujan $$\Xi$$-function given by Ki (Ramanujan J 17(1):123–143, 2008), and we provide new proofs of his results. Our approach is motivated by the ideas of Velásquez (J Anal Math 110:67–127, 2010) in the study of the zeros of certain sums of entire functions with some condition of stability related

更新日期：2020-08-29
• Ramanujan J. (IF 0.79) Pub Date : 2020-08-04
Kazuhide Matsuda

In this paper, we focus on applications of high-level versions of Jacobi’s derivative formula to number theory, such as quaternary quadratic forms and convolution sums of some arithmetical functions.

更新日期：2020-08-05
• Ramanujan J. (IF 0.79) Pub Date : 2020-08-04
Wei Zhang

Let $$\lambda _{f}(n)$$ be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with $$\lambda _{f}(n)$$ over some special sequences. More precisely, for any fixed integer $$k\ge 2$$ and $$j\in \{1,2,3,4\},$$ we are interested in the following sums\begin{aligned} S_{k}(x,j):=\sum _{n\le x}\lambda _{k,f}(n^{j})=\sum _{n\le x}\sum 更新日期：2020-08-04 • Ramanujan J. (IF 0.79) Pub Date : 2020-08-03 Ardavan Afshar We investigate the analogues, in $$\mathbb {F}_q[t]$$, of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maximum of the divisor function at every degree up to an error of a constant, which is significantly smaller 更新日期：2020-08-03 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-30 Igoris Belovas We consider the absolute values of the modified Selberg zeta-function at places symmetric with respect to the critical line. We prove an inequality for the modified Selberg zeta-function in a different way, reproving and extending the result of Garunkštis and Grigutis and completing the extension of a result of Belovas and Sakalauskas. 更新日期：2020-07-30 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-29 Arindam Biswas, Jyoti Prakash Saha Let $$W,W'\subseteq G$$ be non-empty subsets in an arbitrary group G. The set $$W'$$ is said to be a complement to W if $$W\cdot W'=G$$ and it is minimal if no proper subset of $$W'$$ is a complement to W. We show that, if W is finite then every complement of W has a minimal complement, answering a problem of Nathanson. This also shows the existence of minimal r-nets for every $$r\geqslant 0$$ in finitely 更新日期：2020-07-30 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-29 Shoyu Nagaoka Following (Nagaoka and Takemori in Ramanujan J 42: 385–400, 2017), we study some congruence properties satisfied by the theta series associated with Niemeier lattices. 更新日期：2020-07-29 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-28 Shane Chern Let s(n) be the number of 1-shell totally symmetric plane partitions (TSPPs) of n. In this paper, an infinite family of congruences modulo powers of 5 for s(n) will be deduced through an elementary approach. Namely,\begin{aligned} s\left( 2\cdot 5^{2\alpha -1}n+5^{2\alpha -1}\right) \equiv 0 \pmod {5^{\alpha }}. \end{aligned}$$更新日期：2020-07-29 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-28 Sheng-Chi Liu, Shenhui Liu Let $$S(t,F):=\pi ^{-1}\arg L\big (\frac{1}{2}+it,F\big ),$$ where F is a Hecke–Maass cusp form for $$\mathrm {SL}_3({\mathbb {Z}}).$$ We establish an asymptotic formula for the spectral moments of S(t, F), and obtain several other results on S(t, F). 更新日期：2020-07-29 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-28 Lenny Jones Let $$D_n$$ denote the dihedral group of order 2n. We find infinite $$D_6$$-families and an infinite $$D_3$$-family of monic irreducible reciprocal sixth-degree polynomials $$f(x)\in \mathbb {Z}[x]$$, such that $$\{1,\theta ,\theta ^2,\theta ^3,\theta ^4,\theta ^5\}$$ is a basis for the ring of integers of $$L=\mathbb {Q}(\theta )$$, where $$f(\theta )=0$$. 更新日期：2020-07-29 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-28 Su-Ping Cui, Nancy S. S. Gu, Chen-Yang Su Ramanujan presented four identities for third-order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. proved these identities by using q-series. In this paper, using some identities for the universal mock theta function$$\begin{aligned} g(x;q)=x^{-1}\left( -1+\sum _{n=0}^{\infty }\frac{q^{n^

更新日期：2020-07-28
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
P. Guerzhoy, Ka Lun Wong

Farkas in (On an Arithmetical Function II. Complex Analysis and Dynamical Systems II. Contemporary Mathematics, American Mathematical Society, Providence, 2005) introduced an arithmetic function $$\delta$$ and found an identity involving $$\delta$$ and a sum of divisor function $$\sigma '$$. The first-named author and Raji in (Ramanujan J 19(1):19–27, 2009) discussed a natural generalization of the

更新日期：2020-07-28
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-28
Xiaodong Zhao

In this paper, we will investigate three Goldbach–Linnik type problems on cubes of primes. For example, it was proved that, for $$k=1364,$$ every pair of sufficiently large even integers can be represented by a pair of eight prime cubes and k powers of 2. In this paper, we give the detailed proof for the first time and sharpen the value of k to 658.

更新日期：2020-07-28
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
Byeong Moon Kim, Ji Young Kim

A positive definite Hermitian lattice is said to be 1-universal if it represents all positive definite unary Hermitian lattices, including both free and non-free Hermitian lattices. This paper is more concerned with the representations of unary non-free Hermitian lattices by Hermitian lattices. We estimate the minimal rank$$u_m^1$$ of 1-universal Hermitian lattices and we classify all 1-universal binary

更新日期：2020-07-28
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
Ryojun Ito

In this paper, we consider L-functions of two modular forms of weight 3, which are products of the Jacobi theta series, and express their special values at $$s=3$$, 4 in terms of special values of Kampé de Fériet hypergeometric functions. Moreover, via L-values, we give some relations between special values of Kampé de Fériet hypergeometric functions and generalized hypergeometric functions.

更新日期：2020-07-28
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
Kui Liu, Xianchang Meng

This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefully about both the “visible” points and the “invisible” points in the definition of previous research. We prove asymptotic formulas for the number of lattice

更新日期：2020-07-28
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
Jinfa Cheng, Weizhong Dai

In this article, we obtain the adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices, and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well. The particular solutions of the adjoint difference equation are then obtained. As an application of these particular solutions, we use them to obtain

更新日期：2020-07-28
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-27
Ji-Ke Ge, Qiu-Ming Luo

In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we also obtain many interesting identities of Jacobi’s theta functions.

更新日期：2020-07-27
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-26
Nicolás Sirolli, Gonzalo Tornaría

We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. This formula extends our previous work, allowing to compute these central values when the main central value vanishes.

更新日期：2020-07-26
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-26
Miguel A. Méndez

By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers–Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers–Ramanujan continued fraction. By specializing the non-commutative series

更新日期：2020-07-26
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-25
Prapanpong Pongsriiam, Robert C. Vaughan

Let $$n, q, a\in \mathbb N$$, d(n) the number of positive divisors of n, and A(x, q, a) the sum of d(n) over $$n\le x$$ and $$n\equiv a\pmod q$$. Previously we obtained an asymptotic formula for the second moment \begin{aligned} V(x,Q) = \sum _{q\le Q}\sum _{a=1}^q \left| A(x,q,a)-M(x,q,a)\right| ^2, \end{aligned} where $$Q\le x$$ and M(x, q, a) is a usual expected value for A(x, q, a). In this

更新日期：2020-07-25
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-24
Chen Wang

In this paper, we mainly prove two conjectural supercongruences of Sun by using the following identity: \begin{aligned} \sum _{k=0}^n\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right) ^2=16^n\sum _{k=0}^n \frac{\left( {\begin{array}{c}n+k\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}2k\\ k\end{array}}\right) 更新日期：2020-07-25 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-24 Noriyuki Otsubo We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the hypergeometric differential equation. Analogy for q-hypergeometric functions is also studied. 更新日期：2020-07-25 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-24 Jean-Baptiste Gramain, Rishi Nath, James A. Sellers A tremendous amount of research has been done in the last two decades on (s, t)-core partitions when s and t are relatively prime integers. Here we change perspective slightly and explore properties of (s, t)-core and $$(\bar{s},\bar{t})$$-core partitions for s and t with a nontrivial common divisor g. We begin by recovering, using the g-core and g-quotient construction, the generating function for 更新日期：2020-07-25 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-24 Jonathan Bloom, Nathan McNew A partition $$\alpha$$ is said to contain another partition (or pattern) $$\mu$$ if the Ferrers board for $$\mu$$ is attainable from $$\alpha$$ under removal of rows and columns. We say $$\alpha$$ avoids $$\mu$$ if it does not contain $$\mu$$. In this paper we count the number of partitions of n avoiding a fixed pattern $$\mu$$, in terms of generating functions and their asymptotic growth rates 更新日期：2020-07-25 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-24 Francesc Bars, Josep González, Xavier Xarles For a square-free integer N, we present a procedure to compute $$\mathbb {Q}$$-curves parametrized by rational points of the modular curve $$X_0^*(N)$$ when this is hyperelliptic. 更新日期：2020-07-24 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-24 Shaoyun Yi We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level $${\mathfrak {p}}^2$$ for all irreducible, admissible representations of $$\mathrm{GSp}(4, F)$$ with trivial central character for F a $${\mathfrak {p}}$$-adic field. 更新日期：2020-07-24 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-24 Zilong He, Ben Kane Inspired by Dickson’s classification of regular diagonal ternary quadratic forms, we prove that there are no primitive regular ternary m-gonal forms when m is sufficiently large. In order to do so, we construct sequences of primes that are inert in a certain quadratic field and show that they satisfy a certain inequality bounding the next such prime by a product of the previous primes, a question of 更新日期：2020-07-24 • Ramanujan J. (IF 0.79) Pub Date : 2020-07-23 Shane Chern Let $$(m_1,\ldots ,m_J)$$ and $$(r_1,\ldots ,r_J)$$ be two sequences of J positive integers satisfying $$1\le r_j< m_j$$ for all $$j=1,\ldots ,J$$. Let $$(\delta _1,\ldots ,\delta _J)$$ be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product\begin{aligned} \prod _{j=1}^J\Bigg (\prod _{k\ge 1}\big (1-q^{r_j+m_j(k-1)}\big

更新日期：2020-07-24
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
K. R. Vasuki, M. V. Yathirajsharma

On page 237–238 of his second notebook, Ramanujan recorded five modular equations of composite degree 25. Berndt proved all these using the method of parametrization. He also expressed that his proofs undoubtedly often stray from the path followed by Ramanujan. The purpose of this paper is to give direct proofs to four of the five modular equations using the identities known to Ramanujan.

更新日期：2020-07-24
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-23

Let $$r\ge 1$$ be an integer and $$\mathbf{U}:=(U_{n})_{n\ge 0}$$ be the Lucas sequence given by $$U_0=0$$, $$U_1=1,$$ and $$U_{n+2}=rU_{n+1}+U_n$$, for all $$n\ge 0$$. In this paper, we show that there are no positive integers $$r\ge 3,~x\ne 2,~n\ge 1$$ such that $$U_n^x+U_{n+1}^x$$ is a member of $$\mathbf{U}$$.

更新日期：2020-07-23
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
K. Castillo, M. N. de Jesus, F. Marcellán, J. Petronilho

In this work we study orthogonal polynomials via polynomial mappings in the framework of the $$H_q$$-semiclassical class. We consider two monic orthogonal polynomial sequences $$\{p_n (x)\}_{n\ge 0}$$ and $$\{q_n(x)\}_{n\ge 0}$$ such that \begin{aligned} p_{kn}(x)=q_n(x^k),\quad n=0,1,2,\ldots , \end{aligned} where $$k\ge 2$$ is a fixed integer number, and we prove that if one of the sequences

更新日期：2020-07-23
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-23
Li-Yuan Wang, Hao Pan

Let p be an odd prime and let $${\mathbb {F}}_p$$ denote the finite field with p elements. Suppose that g is a primitive root of $${\mathbb {F}}_p$$. Define the permutation $$\tau _g:\,{\mathcal {H}}_p\rightarrow {\mathcal {H}}_p$$ by \begin{aligned} \tau _g(b):={\left\{ \begin{array}{ll} g^b&{}\text {if }g^b\in {\mathcal {H}}_p,\\ -g^b&{}\text {if }g^b\not \in {\mathcal {H}}_p,\\ \end{array}\right

更新日期：2020-07-23
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-22
Jolanta Marzec

It is known that among Siegel modular forms of degree 2 and level 1 the only functions that violate the Ramanujan conjecture are Saito–Kurokawa lifts of modular forms of level 1. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for $$\mathrm {GSp}_4$$ is predicted to fail only in case of CAP representations. It is not known

更新日期：2020-07-23
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-22
James Mc Laughlin

Motivated by results of Hirschhorn, Tang, and Baruah and Kaur on vanishing coefficients (in arithmetic progressions) in a new class of infinite product which have appeared recently, we further examine such infinite products, and find that many such results on vanishing coefficients may be grouped into families. For example, one result proved in the present paper is that if $$b\in \{1,2,\dots , 9, 10\}$$

更新日期：2020-07-23
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-22
A. V. Kisselev

The Hankel transform $$\mathcal {H}_n [f(x)](q)=\int _0^{\infty } \!\! \, x f(x) J_n(q x) \mathrm{d}x$$ is studied for integer $$n\geqslant -1$$ and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of q, provided special conditions on the function f(x) and its derivatives are imposed. It is necessary to underline

更新日期：2020-07-22
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
Biao Wang

In this short note, we show an analogue of one of Alladi’s and Dawsey’s formulas with respect to the Ramanujan sum $$c_n(m)$$ for $$m\geqslant 1$$. Their formulas may be viewed as the case $$m=1$$ in our result.

更新日期：2020-07-22
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
Michael Chou, Pete L. Clark, Marko Milosevic

Let F be a number field, and let $$F^{{\text {cyc}}}$$ be obtained by adjoining to F all the roots of unity. We show that as E ranges over all elliptic curves defined over $$F^{{\text {cyc}}}$$ with complex multiplication, the torsion subgroups $$E(F^{{\text {cyc}}})$$ are finite and uniformly bounded in size.

更新日期：2020-07-22
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
Doris D. M. Sang, Diane Y. H. Shi

In 2013, Bringmann and Mahlburg defined a new type of partitions by adding a different restriction on the smallest parts in Gleissberg’s generalization of partitions considered by Schur. The generating function of this new Schur-type partitions is a mixed mock modular form, more precisely, it equals the product of the generating function of Gleissberg’s generalization and a specialization of a universal

更新日期：2020-07-22
• Ramanujan J. (IF 0.79) Pub Date : 2020-07-21
Olga Balkanova

We prove an asymptotic formula for the twisted first moment of Maaß form symmetric square L-functions on the critical line and at the central point. The error term is estimated uniformly with respect to all parameters.

更新日期：2020-07-22
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