The Ramanujan Journal最新文献

The problem of the integrability of Mellin transforms is presented. Sufficient Lipschitz conditions are given to solve this problem. These results are inspired by well-known works of Titchmarsh in classical Fourier harmonic analysis. Some results on the integrability of Mellin transforms of the Mellin convolutions are also given.

We study the Iwasawa theory of p-primary Selmer groups of elliptic curves E over a number field K. Assume that E has additive reduction at the primes of K above p. In this context, we prove that the Iwasawa invariants satisfy an analogue of the Riemann–Hurwitz formula. This generalizes a result of Hachimori and Matsuno. We apply our results to study rank stability questions for elliptic curves in prime cyclic extensions of (mathbb {Q}). These extensions are ordered by their absolute discriminant and we prove an asymptotic lower bound for the density of extensions in which the Iwasawa invariants as well as the rank of the elliptic curve is stable.

Let m be a positive integer, ((w_n)) be a sequence of positive integers, and ((y_n)) be a sequence of nonzero integers with (y_1ge 1). Define (q_0=1, q_1=w_0, q_{n+1}=q_{n-1}(w_nq_n^m+y_n) ,,(nge 1)). Under certain assumptions on ((w_n)) and ((y_n)), we give the exact value of the irrationality exponent of the number

Let n, p and j be integers. Define

The coefficients of the polynomial (R_{n,p,j}(q)) count certain regular partition. Recently, Dong and Ji studied unimodality of the polynomials (R_{n,p,p-1}(q)). As an extension, in this paper, we give a criterion for unimodality of the polynomials ( R_{n,p,j}(q)) for (p ge 6) and (lceil frac{p+1}{2}rceil le jle p-1.) In particular, using our criterion and Mathematica, we obtain that (R_{n,p,j}(q)) is unimodal for (nge 3) if (6le p le 15) and (lceil frac{p+1}{2}rceil le jle p-1.)

Recently, Andrews and Paule introduced a partition function PDN1(N) which counts the number of partition diamonds with (n+1) copies of n where summing the parts at the links gives N. They also established the generating function of PDN1(n) and proved congruences modulo 5,7,25,49 for PDN1(n). At the end of their paper, Andrews and Paule asked for the existence of other types of congruence relations for PDN1(n). Motivated by their work, we prove some new congruences modulo 125 and 625 for PDN1(n) by using some identities due to Chern and Tang. In particular, we discover a family of strange congruences modulo 625 for PDN1(n). For example, we prove that for (kge 0),

We describe the p-divisibility transposition for the Fourier coefficients of Hermitian modular forms. The results show that the same phenomenon as that for Siegel modular forms holds for Hermitian modular forms

Hecke-type double sums play a crucial role in proving many identities related to mock theta functions given by Ramanujan. In the literature, the Bailey pair machinery is an efficient tool to derive Hecke-type double sums for mock theta functions. In this paper, by using some Bailey pairs and conjugate Bailey pairs, and then applying the Bailey transform, we establish some trivariate identities which imply the Hecke-type double sums for some classical mock theta functions of orders 3, 6, and 10. Meanwhile, we generalize a bivariate Hecke-type identity due to Garvan.

In this paper, we describe a systematic way of obtaining the exact generating functions for (overline{p}(2n)), (overline{p}(4n)) (first proved by Fortin et al.), (overline{p}(8n)), (overline{p}(16n)), etc. where (overline{p}(n)) denotes the number of overpartitions of n. We further establish several new infinite families of congruences modulo (2^4) and (2^5) for (overline{p}(n)). For example, we prove that for all (n, alpha , beta ge 0) and primes (pge 5),

and

where (bigl (frac{-6}{p}bigr )=-1) and (1le jle p-1). The last congruence was proved by Xiong (Int J Number Theory 12:1195–1208, 2016) for modulo (2^4).

There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter (r+1). Motivated by work on rational numbers in the Calkin–Wilf tree, we classify partitions whose corresponding binary sequence is a palindrome. We give a generating function that counts these partitions, and describe how to efficiently generate all of them. Atypically for partition generating functions, we find an unusual significance to prime degrees. Specifically, we prove there are nontrivial palindrome partitions of n except when (n=3) or (n+1) is prime. We find an interesting new “branching diagram” for partitions, similar to Young’s lattice, with an action of the Klein four group corresponding to natural operations on the binary sequences.

We examine the family of generalized Laguerre polynomials (L_{n}^{(n)}(x)). In 1989, Gow discovered that if n is even, then the discriminant of (L_{n}^{(n)}(x)) is a nonzero square of a rational number. Additionally, in the case where the polynomial (L_{n}^{(n)}(x)) is irreducible over the rationals, the associated Galois group is the alternating group (A_{n}). Filaseta et al. (2012) established the irreducibility of (L_{n}^{(n)}(x)) for every (n>2) satisfying (2pmod {4}). They also demonstrated that if n is (0pmod {4}), then (L_{n}^{(n)}(x)) has a linear factor if it is not irreducible. The question of whether (L_{n}^{(n)}(x)) has a linear factor when n is (0pmod {4}) remained unanswered. We resolve this question by proving that (L_{n}^{(n)}(x)) does not have a linear factor for sufficiently large n. This conclusion completes the classification of generalized Laguerre polynomials having Galois group the alternating group, excluding a finite set of exceptions.