Showing papers in "Ramanujan Journal in 2016"

Hypergeometric series, modular linear differential equations and vector-valued modular forms

Abstract: We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.

Updating the error term in the prime number theorem

Abstract: An improved estimate is given for $$|\theta (x) -x|$$ , where $$\theta (x) = \sum _{p\le x} \log p$$ . Four applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, the third to a conjecture by Pomerance and the fourth to an inequality studied by Ramanujan.

Partitions, quasimodular forms, and the Bloch–Okounkov theorem

Abstract: We give a very short proof of the Bloch–Okounkov theorem on the quasimodularity of certain functions defined by sums over partitions, and also show how to make their map \(\mathfrak {s}\mathfrak {l}_2\)-equivariant.

The 3D index of an ideal triangulation and angle structures

Abstract: The 3D index of Dimofte–Gaiotto–Gukov is a partially defined function on the set of ideal triangulations of 3-manifolds with r tori boundary components For a fixed 2r tuple of integers, the index takes values in the set of q-series with integer coefficients Our goal is to give an axiomatic definition of the tetrahedron index and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure The latter is a generalization of a strict angle structure We also prove that the 3D index is invariant under 3–2 moves, but not in general under 2–3 moves

The tail of a quantum spin network

Abstract: The tail of a sequence $$\{P_n(q)\}_{n \in \mathbb {N}}$$ of formal power series in $$\mathbb {Z}[[q]]$$ is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of $$P_n$$ . This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews–Gordon identities for the two-variable Ramanujan theta function as well to corresponding identities for the false theta function.

The Frobenius problem for repunit numerical semigroups

Abstract: A repunit is a number consisting of copies of the single digit 1. The set of repunits in base b is \(\big \{\frac{b^n-1}{b-1} ~|~ n\in {\mathbb N}\backslash \{0\}\big \}\). A numerical semigroup S is a repunit numerical semigroup if there exist integers \(b\in {\mathbb N}\backslash \left\{ 0,1\right\} \) and \(n\in {\mathbb N}\backslash \left\{ 0\right\} \) such that \(S=\big \langle \big \{\frac{b^{n+i}-1}{b-1} ~|~ i\in {\mathbb N}\big \}\big \rangle \). In this work, we give formulas for the embedding dimension, the Frobenius number, the type and the genus for a repunit numerical semigroup.

Regularized inner products of modular functions

Abstract: In this note, we give an explicit basis for the harmonic weak forms of weight two. We also show that their holomorphic coefficients can be given in terms of regularized inner products of weight zero weakly holomorphic forms.

The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

Abstract: We study the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in $${\mathbb {Q}}$$ arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . The (quasi-)modular forms for the full modular group $${{\mathrm{SL}}}_2({\mathbb {Z}})$$ constitute a subalgebra of $${{\mathrm{{\mathcal {MD}}}}}$$ , and this also yields linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those of length 2, coming from modular forms.

Parité des coefficients de formes modulaires

Abstract: Soit \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) la reduction modulo 2 de la fonction \(\Delta \). Une forme modulaire de niveau \(1\), \(f=\sum _{n\geqslant 0} c(n) \,q^n\), a coefficients entiers, est congrue modulo \(2\) a un polynome en \(\Delta \). Soit \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { impair }} 1\) le nombre de coefficients de Fourier impairs de \(f\) d’indice \(\leqslant x\). L’ordre de grandeur de \(W_f(x)\) a ete determine par Serre dans les annees 70. Nous donnons ici un equivalent de \(W_f(x)\). Soit \(p(n)\) la fonction de partition et \(A_0(x)\) (resp. \(A_1(x)\)) le nombre d’entiers \(n \leqslant x\) tels que \(p(n)\) est pair (resp. impair). Dans des articles anterieurs, J.-L. Nicolas a montre que \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) pour \(x\geqslant 3\) et que \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) pour \(x\geqslant 7\). On prouve ici que \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) pour \(x>1\) et que \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) pour \(x\geqslant 2\). Les principaux outils utilises dans la preuve de ces resultats sont la determination de l’ordre de nilpotence d’une forme modulaire de niveau 1 modulo 2, et de la structure de l’espace de ces formes modulaires en tant que module sur l’algebre de Hecke, obtenus dans un travail recent de J.-L. Nicolas et J.-P. Serre.

On some congruences of certain binomial sums

Abstract: For any prime \(p>3,\) we prove that $$\begin{aligned} \sum _{k=0}^{p-1}\frac{3k+1}{(-8)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-1}{p}\right) +p^3E_{p-3}\pmod {p^4}, \end{aligned}$$ where \(E_{0},E_{1},E_{2},\ldots \) are Euler numbers and \(\left( \frac{\cdot }{p}\right) \) is the Legendre symbol. This result confirms a conjecture of Z.-W. Sun. We also re-prove that for any odd prime \(p,\) $$\begin{aligned} \sum _{k=0}^{\frac{p-1}{2}}\frac{6k+1}{(-512)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-2}{p}\right) \pmod {p^2} \end{aligned}$$ using WZ method.

An infinite family of congruences modulo $$$$ for $$1$$1-regular bipartitions

Abstract: Let \(B_{13}(n)\) denote the number of \(13\)-regular bipartitions of \(n\). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function \(p(n)\). In particular, we shall prove an infinite family of congruences: for \(\alpha \ge 2\) and \(n\ge 0\), $$\begin{aligned} B_{13}(3^{\alpha }n+2\cdot 3^{\alpha -1}-1)\equiv \ 0\ (\mathrm{mod\ }3). \end{aligned}$$ In addition, we will also give an alternative proof of one infinite family of congruences for \(b_{13}(n)\), the number of \(13\) regular partitions of \(n\), due to Webb.

Restricted k-color partitions

Abstract: We generalize overpartitions to (k, j)-colored partitions: k-colored partitions in which each part size may have at most j colors We find numerous congruences and other symmetries We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps In the process of proving certain congruences, we find results of independent interest on the number of partitions with exactly 2 sizes of part in several arithmetic progressions We find connections to divisor sums, the Han/Nekrasov–Okounkov hook length formula and a possible approach to finitization, and other topics, suggesting that a rich mine of results is available We pose several immediate questions and conjectures

Polyharmonic Maass forms for PSL(2,Z)

Abstract: We discuss the space of polyharmonic Maass forms of even integer weight on $$\text {PSL}(2,\mathbb Z)\backslash \mathbb H$$ . We explain the role of the real-analytic Eisenstein series $$E_k(z,s)$$ and the differential operator $$\frac{\partial }{\partial s}$$ in this theory.

Analogues of Jacobi's derivative formula

Abstract: In this work, we use Jacobi’s derivative formula to obtain analogues to theta constants with rational characteristics. Furthermore, we show that the analogues yield many product-series identities by only using the elementary fact that a holomorphic elliptic function is a constant.

Homogeneous q-difference equations and generating functions for q-hypergeometric polynomials

Abstract: In this paper we show how to deduce several types of generating functions for $$q$$ -hypergeometric polynomials by the method of homogeneous $$q$$ -difference equations. In addition, we build relations between transformation formulas and homogeneous $$q$$ -difference equations. Moreover, we generalize the Andrews–Askey integral from the perspective of $$q$$ -integrals by the method of homogeneous $$q$$ -difference equations.

Congruences involving g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k

Abstract: Define $$g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$ for $$n=0,1,2,\ldots $$ . Those numbers $$g_n=g_n(1)$$ are closely related to Apery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $$g_n(x)$$ . For example, for any prime $$p>5$$ we have $$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$ This is similar to Wolstenholme’s classical congruences $$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$ for any prime $$p>3$$ .

Mehler–Heine type formulas for Charlier and Meixner polynomials

Abstract: We derive Mehler–Heine type asymptotic formulas for Charlier and Meixner polynomials, and also for their associated families. These formulas provide good approximations for the polynomials in the neighborhood of $$x=0$$ and determine the asymptotic limit of their zeros as the degree $$n$$ goes to infinity.

Some congruences for 6-colored generalized Frobenius partitions

Abstract: We prove some congruences discovered by Baruah and Sarmah and by Xia for $$c\phi _6(n)$$ , the number of 6-colored generalized Frobenius partitions of n.

Integrals associated with Ramanujan and elliptic functions

Abstract: We evaluate in closed form certain classes of infinite integrals containing trigonometric and hyperbolic trigonometric functions in their integrands. Although not apparent, the integrals are related to Jacobian elliptic functions, in particular, to theorems found in Ramanujan’s notebooks. One of our evaluations answers a question first posed by M. E. H. Ismail.

Interlacing properties and bounds for zeros of $${}_2\phi _1$$ hypergeometric and little q-Jacobi polynomials

Abstract: Let $$\displaystyle \{p_n\}_{n=0}^{\infty }$$ , where $$p_n$$ is a polynomial of degree n, be a sequence of polynomials orthogonal with respect to a positive probability measure If $$x_{1,n} < \cdots < x_{n,n}$$ denotes the zeros of $$p_n$$ while $$x_{1,n-1} < \cdots < x_{n-1,n-1}$$ are the zeros of $$p_{n-1}$$ , the inequality $$\begin{aligned} x_{1,n} < x_{1,n-1} < x_{2,n} < \cdots < x_{n-1,n}< x_{n-1,n-1}< x_{n,n}, \end{aligned}$$ known as the interlacing property, is satisfied We use a consequence of a generalised version of Markov’s monotonicity results to investigate interlacing properties of zeros of contiguous basic hypergeometric polynomials associated with little q-Jacobi polynomials and determine inequalities for extreme zeros of the above two polynomials It is observed that the new bounds which are obtained in this paper give more precise upper bounds for the smallest zero of little q-Jacobi polynomials, improving previously known results by Driver and Jordaan (Math Model Nat Phenom 8(1):48–59, 2013), and in some cases, those by Gupta and Muldoon (J Inequal Pure Appl Math 8(1):7, 2007) Numerical examples are given in order to illustrate the accuracy of our bounds

Linearization formulae for certain Jacobi polynomials

Abstract: In this article, some new linearization formulae of products of Jacobi polynomials for certain parameters are derived. These new derived formulae are expressed in terms of hypergeometric functions of unit argument, and they generalize some existing formulae in the literature. With the aid of some standard formulae and also by employing symbolic algebraic computation, and in particular Zeilberger’s algorithm, several reduction formulae for summing certain terminating hypergeometric functions of unit argument are given, and hence several linearization formulae of products of Jacobi polynomials for special parameters free of hypergeometric functions are deduced.

Family complexity and cross-correlation measure for families of binary sequences

Abstract: We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al. in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family \((e_{i,1},\ldots ,e_{i,N})\in \{-1,+1\}^N\), \(i=1,\ldots ,F\), of binary sequences of length \(N\) in terms of the cross-correlation measure of its dual family \((e_{1,n},\ldots ,e_{F,n})\in \{-1,+1\}^F\), \(n=1,\ldots ,N\). We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo \(p\) with middle coefficient \(0\), that is, \(e_{i,n}=\big (\frac{n^2-bi^2}{p}\big )_{n=1}^{(p-1)/2}\) for \(i=1,\ldots ,(p-1)/2\), where \(b\) is a quadratic nonresidue modulo \(p\), showing that this family as well as its dual family has both a large family complexity and a small cross-correlation measure up to a rather large order.

Two operator representations for the trivariate q -polynomials and Hahn polynomials

Abstract: In this paper, we introduce a trivariate q-polynomials $$F_n(x,y,z;q)$$ as a general form of Hahn polynomials $$\psi _n^{(a)}(x|q)$$ and $$\psi _n^{(a)}(x,y|q)$$ . We represent $$F_n(x,y,z;q)$$ by two operators: the homogeneous q-shift operator $$L(b\theta _{xy})$$ given by Saad and Sukhi (Appl Math Comput 215:4332–4339, 2010), and the Cauchy companion operator $$E(a,b;\theta )$$ given by Chen (q-Difference Operator and Basic Hypergeometric Series, 2009) to derive the generating function, symmetric property, Mehler’s formula, Rogers formula, another Roger-type formula, linearization formula, and an extended Rogers formula for the trivariate q-polynomials. Then, we give the corresponding formulas for our new definitions of Hahn polynomials $$\psi _n^{(a)}(x|q)$$ and $$\psi _n^{(a)}(x,y|q)$$ by representing Hahn polynomials by the operators $$L(b\theta _{xy})$$ and $$E(a,b;\theta )$$ , and by a special substitution in the trivariate q-polynomials $$F_n(x,y,z;q)$$ .

On the coefficients of asymptotic expansion for the harmonic number by Ramanujan

Abstract: Let $$H_n=\sum _{k=1}^{n}\frac{1}{k}$$ be the $$n$$ th harmonic number In this paper, we establish a new asymptotic expansion of $$H_n$$ By using the result obtained, we derive a recurrence relation for determining the coefficients of Ramanujan’s asymptotic expansion for the harmonic number Also, we establish asymptotic expansion of $$H_n$$ in terms of $$n(n+1)+1/3$$

On generalized Ramanujan primes

Abstract: In this paper, we establish several results concerning the generalized Ramanujan primes. For $$n\in \mathbb {N}$$ and $$k \in \mathbb {R}_{> 1}$$ , we give estimates for the $$n$$ th $$k$$ -Ramanujan prime, which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of $$k$$ -Ramanujan primes. In addition, we find explicit formulae for certain $$n$$ th $$k$$ -Ramanujan primes. As an application, we prove that a conjecture of Mitra et al. ( arXiv:0906.0104v1 , 2009) concerning the number of primes in certain intervals holds for every sufficiently large positive integer.

Series representations for special functions and mathematical constants

Abstract: We develop a collection of single-parameter series representations involving special functions and mathematical constants The techniques used result in new representations as well as alternative proofs of known representations

Integrals of K and E from lattice sums

Abstract: We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.

Congruences for (3,11)-regular bipartitions modulo 11

Abstract: In this note we investigate the function Bk,� (n), which counts the number of (k ,� ) -regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for α ≥ 2 and n ≥ 0, B3,11 3 α n + 5 · 3 α−1 − 1 2 ≡ 0 (mod 11).

Shifted convolution of cusp-forms with $$\theta $$-series

Abstract: In this paper we apply some simple approaches to improve a recent result of Luo on the shifted convolution sum of the Fourier coefficients of a cusp form and a theta series. A mean square formula is established in order to explore the right order of magnitude of this shifted convolution sum.

Generators and relations of the graded algebra of modular forms

Abstract: We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in \(\mathbb {Q}\) of level \(\Gamma _{0}(N)\) for \(N\) satisfying some congruence conditions, and of level \(\Gamma _{1}(N)\). We give similar bounds for the graded \(\mathbb {Z}[\frac{1}{N}]\)-algebra of modular forms of level \(\Gamma _{1}(N)\) with coefficients in \(\mathbb {Z}[\frac{1}{N}]\). For a prime \(p \ge 5\), we give a lower bound on the highest weight appearing in a minimal list of generators for \(\Gamma _{0}(p)\), and we identify a set of generators for the graded algebra \(M(\Gamma _{0}(p),\mathbb {Z})\) of modular forms of level \(\Gamma _{0}(p)\) with coefficients in \(\mathbb {Z}\), showing that, in contrast to the cases studied in the study of Rustom (J. Number Theory 138:97–118, 2014), this weight is unbounded. We generalize a result of Serre concerning congruences between modular forms of level \(\Gamma _{0}(p)\) and \(SL_2(\mathbb {Z})\), and use it to identify a set of generators for \(M(\Gamma _{0}(p),\mathbb {Z})\), and we state two conjectures detailing further the structure of this algebra. Finally, we provide computations concerning the number of generators and relations for each of these algebras, as well as computational evidence for these conjectures.