The sixth moment of the family of Γ<Subscript>1</Subscript>(<Emphasis Type="Italic">q</Emphasis>)-automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper we investigate the sixth moment of the family of L-functions associated to holomorphic modular forms on GL ₂ with respect to a congruence subgroup Γ₁(q). The bound for central values averaged over the family, consistent with the Lindelöf hypothesis, is obtained for prime levels q.     

Uniform subconvexity and symmetry breaking reciprocity

A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the ?-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level ? and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.     

A q-summation formula,the continuous q-Hahn polynomials and the big q-Jacobi polynomials

Using a general q-summation formula, we derive a generating function for the q-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the continuous q-Hahn polynomials. A new proof of the orthogonality relation for the big q-Jacobi polynomials is also given. A simple evaluation of the Nassrallah–Rahman integral is derived by using this summation formula. A new q-beta integral formula is established, which includes the Nassrallah–Rahman integral as a special case. The q-summation formula also allows us to recover several strange q-series identities.     

A q-series expansion formula and the Askey–Wilson polynomials

Previously, we proved a q-series expansion formula which allows us to recover many important classical results for q-series. Based on this formula, we derive a new q-formula in this paper, which clearly includes infinitely many q-identities. This new formula is used to give a new proof of the orthogonality relation for the Askey–Wilson polynomials. A curious q-transformation formula is proved, and many applications of this transformation to Hecke type series are given. Some Lambert series identities are also derived.     

q-Bernoulli polynomials and q-umbral calculus

In this paper, we investigate some properties of q-Bernoulli polynomials arising from q-umbral calculus. We find a formula for expressing any polynomial as a linear combination of q-Bernoulli polynomials with explicit coefficients. Also, we establish some connections between q-Bernoulli polynomials and higher-order q-Bernoulli polynomials.     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

A q-Analogue of the Stirling Formula and a Continuous Limiting Behaviour of the q-Binomial Distribution—Numerical Calculations

In this article, we derive an asymptotic formula for the q-factorial number of order n using the saddle point method. This formula is a q-analogue, for 0?<?q?<?1, of the usual Stirling formula for the factorial number of order n. Also, this formula is used to provide a continuous limiting behaviour of the q-Binomial distribution in the sense of pointwise convergence. Specifically, the q-Binomial distribution converges to a continuous Stieltjes–Wigert distribution. Furthermore, we present some numerical calculations, using the computer program MAPLE, indicating a quite strong convergence.     

On the rate of approximation by q modified Beta operators

In the present paper we propose the q analogue of the modified Beta operators. We apply q-derivatives to obtain the central moments of the discrete q-Beta operators. A direct result in terms of modulus of continuity for the q operators is also established. We have also used the properties of q integral to establish the recurrence formula for the moments of q analogue of the modified Beta operators. We also establish an asymptotic formula. In the end we have also present the modification of such q operators so as to have better estimate.     

Orbits of rational n-sets of projective spaces under the action of the linear group

Let k=Fq be a finite field. We enumerate k-rational n-sets of (unordered) points in a projective space PN over k, and we compute the generating function for the numbers of PGLN+1(k)-orbits of these n-sets. For N=1,2 we obtain a formula for these numbers of orbits as a polynomial in q with integer coefficients.     

Two new triangles of q-integers via q-Eulerian polynomials of type A and B

The classical Eulerian polynomials can be expanded in the basis t ^k?1(1+t) ^n+1?2k (1≤k≤?(n+1)/2?) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a q-analogue of this expansion for Carlitz’s q-Eulerian polynomials as well as a similar formula for Chow–Gessel’s q-Eulerian polynomials of type B. We shall give some applications of these two formulas, which involve two new sequences of polynomials in the variable q with positive integral coefficients. It is an open problem to give a combinatorial interpretation for these polynomials.     

The Moment Problem Associated with the q -Laguerre Polynomials

Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.     

Functional definitions for q-analogues of Eulerian functions and applications

We explore a number of functional properties of the q-gamma function and a class of its quotients; including the q-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions on their sign. We prove how these and other functional properties, such as the multiplication formula or asymptotic expansion, together with the fundamental functional equation of the q-gamma function uniquely define those functions. We also study reciprocal “relatives” of the fundamental q-gamma functional equation, and prove uniqueness of solution results for them. In addition, we also use a reflection formula of Askey to derive expressions relating the classical sine function and the number π to the q-gamma function. Throughout we highlight the similarities and differences between the cases 0 < q < 1 and q > 1.     

An analogue of the Harer-Zagier formula for unicellular maps on general surfaces

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is simply connected. In a famous article, Harer and Zagier established a formula for the generating function of unicellular maps counted according to the number of vertices and edges. The keystone of their approach is a counting formula for unicellular maps on orientable surfaces with n edges, and with vertices colored using every color in [q] (adjacent vertices are authorized to have the same color). We give an analogue of this formula for general (locally orientable) surfaces.Our approach is bijective and is inspired by Lass?s proof of the Harer-Zagier formula. We first revisit Lass?s proof and twist it into a bijection between unicellular maps on orientable surfaces with vertices colored using every color in [q], and maps with vertex set [q] on orientable surfaces with a marked spanning tree. The bijection immediately implies Harer-Zagier?s formula and a formula by Jackson concerning bipartite unicellular maps. It also shed a new light on constructions by Goulden and Nica, Schaeffer and Vassilieva, and Morales and Vassilieva. We then extend the bijection to general surfaces and obtain a correspondence between unicellular maps on general surfaces with vertices colored using every color in [q], and maps on orientable surfaces with vertex set [q]with a marked planar submap. This correspondence gives an analogue of the Harer-Zagier formula for general surfaces. We also show that this formula implies a recursion formula due to Ledoux for the numbers of unicellular maps with given numbers of vertices and edges.     

Low moments of Dirichlet series

We determine the maximum possible size of the q ^th moment of a Dirichlet series, for 1 ≦ q≦ 2.     

Orthogonal polynomials and laurent polynomials related to the Hahn-Extonq-Bessel function

Laurent polynomials related to the Hahn-Extonq-Bessel function, which areq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurentq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials, the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurentq-Lommel polynomials and related functions is used to obtain estimates for the latter.     

Two matrix inversions associated with the Hagen-Rothe formula, their q-analogues and applications

By means of the Hagen-Rothe formula, we establish two new matrix inversions with four parameters. These new inversions uniformize Riordan's inverse relations of Abel-, Chebyshev-, and Legendre-type as well as Gould's inversions based on Vandermonde-type convolutions. Some related q-series inverse relations using the known q-analogues of the Hagen-Rothe formula are established. A Λ-extension of Gould's g-inverse, a novel expression for all Chebyshev-type inversions, and several new summation and transformation formulas of series are presented as applications.     

Partial q-analogues of the Pfaff/Cauchy derivative formula

The Pfaff/Cauchy derivative identities are generalizations of Leibniz formula for the nth derivative of a product of two functions. In this paper, we first derive three generalized forms of the q-Leibniz formula. The results are also partial q-analogues of the Pfaff/Cauchy derivative formulae. Then we give some applications and several q-identities are obtained.     

The eighth moment of Dirichlet L-functions

We prove an asymptotic for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q  , over all moduli q?Q$q?Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2)$\mathit{GL}(2)$L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24 024 appears as a factor in the leading order term of the eighth moment.     

Some approximation properties of q-Baskakov-Durrmeyer operators

Very recently Aral and Gupta [1] introduced q analogue of Baskakov-Durrmeyer operators. In the present paper we extend the studies, we establish the recurrence relations for the central moments and obtain an asymptotic formula. Also in the end we propose modified q-Baskakov-Durrmeyer operators, from which one can obtain better approximation results over compact interval.