The Ramanujan Journal最新文献

We introduce a certain discrete probability distribution (P_{n,m,k,l;q}) having non-negative integer parameters n, m, k, l and quantum parameter q which arises from a zonal spherical function of the Grassmannian over the finite field (mathbb {F}_q) with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single q-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating ({}_4 phi _3)-hypergeometric series.

For a prime (pequiv 3) ((text {mod }4)), let (h(-8p)) and h(8p) be the class numbers of (mathbb {Q}(sqrt{-2p})) and (mathbb {Q}(sqrt{2p})), respectively. Let (Psi (xi )) be the Hirzebruch sum of a quadratic irrational (xi ). We show that (h(-8p)equiv h(8p)Big (Psi (2sqrt{2p})/3-Psi (frac{1+sqrt{2p}}{2})/3Big )) ((text {mod }16)). Also, we show that (h(-8p)equiv 2,h(8p)Psi (2sqrt{2p})/3) ((text {mod }8)) if (pequiv 3) ((text {mod }8)), and (h(-8p)equiv big (2,h(8p)Psi (2sqrt{2p})/3big )+4) ((text {mod }8)) if (pequiv 7) ((text {mod }8)).

The main purpose of this article is to study higher order moments of Kummer sums weighted by L-functions using estimates for character sums and analytic methods. The results of this article complement a conjecture of Zhang Wenpeng (2002). Also the results in this article give analogous results of Kummer’s conjecture (1846).

In this paper, we consider the Atkin-like polynomials that appeared in the study of normalized extremal quasimodular forms of depth 1 on (SL_{2}(mathbb {Z})) by Kaneko and Koike as orthogonal polynomials and clarify their properties. Using them, we show that the normalized extremal quasimodular forms have a certain expression by the linear functional corresponding to the Atkin inner product and prove an unexpected connection between generalized Faber polynomials, which are closely related to certain bases of the vector space of weakly holomorphic modular forms, and normalized extremal quasimodular forms. In particular, we reveal that the orthogonal polynomial expansion coefficients of the generalized Faber polynomials by the Atkin-like polynomials appear in the Fourier coefficients of normalized extremal quasimodular forms multiplied by certain (weakly) holomorphic modular forms.

In this paper we investigate a class of terminating (_4phi _3)-series. By utilizing the Carlitz inversion and four contiguous relations for the (Omega _{lambda ,mu })-series, many new summation formulas for terminating (_4phi _3)-series are obtained.

In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function (|Delta (x+iy)|^2), i.e., the Lambert series (sum _{n=1}^infty tau (n)^2 e^{-4 pi n y}), can be expressed in terms of the non-trivial zeros of the Riemann zeta function. In this article, we study an asymptotic expansion of a generalized version of the aforementioned Lambert series associated with Siegel cusp forms.

We study algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) with respect to a weight matrix of the form (W^{(nu )}_{phi }(x) = x^{nu }e^{-phi (x)} W^{(nu )}_textrm{pol}(x)), where (nu >0), (W^{(nu )}_textrm{pol}(x)) is a certain matrix valued polynomial and (phi ) is an analytic function. We introduce differential operators ({mathcal {D}}), ({mathcal {D}}^{dagger }) which are mutually adjoint with respect to the matrix inner product induced by (W^{(nu )}_{phi }(x)). We prove that the Lie algebra generated by ({mathcal {D}}) and ({mathcal {D}}^{dagger }) is finite dimensional if and only if (phi ) is a polynomial. For a polynomial (phi ), we describe the structure of this Lie algebra. As a byproduct, we give a partial answer to a problem by Ismail about finite dimensional Lie algebras related to scalar Laguerre type polynomials. The case (phi (x)=x) is discussed in detail.

Let (mathbb {F}_q[t]) be the polynomial ring over the finite field (mathbb {F}_q) of q elements. For a natural number (Nge 1,) let (mathbb {G}_N) be the subset of (mathbb {F}_q[t]) containing all polynomials of degree less than N. Let (hin mathbb {F}_q[t][x]) be a polynomial of degree (2le k<p,) the characteristic of (mathbb {F}_q.) Suppose that for every (din mathbb {F}_q[t]setminus {0},) there exists (min mathbb {F}_q[t]) such that (dmid h(m)) and ((d,m)=1.) Let (Asubseteq mathbb {G}_N) with (|A|=delta q^N.) Suppose further that ((A-A)cap left( h(Omega )setminus {0}right) =emptyset ,) where (A-A) is the difference set of A and (Omega ) denotes the set of all monic irreducible polynomials in (mathbb {F}_q[t]). It is proved that (delta ll N^{-mu }) for any (0<mu <1/(2k-2),) where the implied constant depends only on (q, h) and (mu .)

We prove that assuming the Generalized Riemann Hypothesis every even integer larger than (exp (exp (14))) can be written as the sum of a prime number and a number that has at most two prime factors.

Munarini (J Integer Seq 23: Article 20.3.8, 2020) recently showed that the derangement polynomial (d_n(q)=sum _{sigma in {mathcal {D}}_n}q^{{{,textrm{maj},}}(sigma )}) is expressible as the determinant of either an (ntimes n) tridiagonal matrix or an (ntimes n) lower Hessenberg matrix. Qi et al. (Cogent Math 3:1232878, 2016) showed that the classical derangement number (d_n=n!sum _{k=0}^nfrac{(-1)^k}{k!}) is expressible as a tridiagonal determinant of order (n+1). We show in this work that similar determinantal expressions exist for the type B derangement polynomial (d_n^B(q)=sum _{sigma in {mathcal {D}}_n^B}q^{{{,textrm{fmaj},}}(sigma )}) studied previously by Chow (Sém Lothar Combin 55:B55b, 2006), and the type D derangement polynomial (d_n^D(q)=sum _{sigma in {mathcal {D}}_n^D}q^{{{,textrm{maj},}}(sigma )}) studied recently by Chow (Taiwanese J Math 27(4):629–646, 2023). Representations of the types B and D derangement numbers (d_n^B) and (d_n^D) as determinants of order (n+1) are also presented.