Instant Evaluation and Demystification of ζ(n),L(n,χ)\zeta(n),L(n,\chi) that Euler,Ramanujan Missed - I

For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive integral argument in first variable.On using functionalequations this results in instant evaluation of Riemann Zeta function at positive even integral values of its argument and of Dirichlet L series at positive integral values of its argument,when the argument and the corresponding Dirichlet character are both even or both odd.We also obtain finite sum expression for any Dirichlet L series,when its argument is one.We also deal with Lerch's Zeta function on similar lines.Comment: 19 page

Relating log-tangent integrals with the Riemann zeta function

We show that integrals involving log-tangent function, with respect to certain square-integrable functions on $(0, \pi/2)$, can be evaluated by some series involving the harmonic number. Then we use this result to establish many closed forms relating to the Riemann zeta function at odd positive integers. In addition, we show that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and that its values depend on the Dirichlet series $\zeta_h(s) :=\sum_{n = 1}^\infty h_n n^{-s}$, where $h_n = \sum_{k=1}^n(2k-1)^{-1}$.Comment: 20 page

Generalized Donaldson-Thomas invariants on the local projective plane

We show that the generating series of generalized Donaldson-Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type series for indefinite lattices. In particular it absolutely converges to give a holomorphic function on the upper half plane.Comment: 25 page

Meixner functions and polynomials related to Lie algebra representations

The decomposition of the tensor product of a positive and a negative discrete series representation of the Lie algebra su(1,1) is a direct integral over the principal unitary series representations. In the decomposition discrete terms can occur, and the discrete terms are a finite number of discrete series representations or one complementary series representation. The interpretation of Meixner functions and polynomials as overlap coefficients in the four classes of representations and the Clebsch-Gordan decomposition, lead to a general bilinear generating function for the Meixner polynomials. Finally, realizing the positive and negative discrete series representations as operators on the spaces of holomorphic and anti-holomorphic functions respectively, a non-symmetric type Poisson kernel is found for the Meixner functions.Comment: 20 page

Ultimate Positivity of Diagonals of Quasi-rational Functions

The problem to decide whether a given multivariate (quasi-)rational function has only positive coefficients in its power series expansion has a long history. It dates back to Szego in 1933 who showed certain quasi-rational function to be positive, in the sense that all the series coefficients are positive, using an involved theory of special functions. In contrast to the simplicity of the statement, the method was surprisingly difficult. This dependency motivated further research for positivity of (quasi-)rational functions. More and more (quasi-)rational functions have been proven to be positive, and some of the proofs are even quite simple. However, there are also others whose positivity are still open conjectures. In this talk, we focus on a less difficult but also interesting question to decide whether the diagonal of a given quasi-rational function is ultimately positive, especially for the one conjectured to be positive by Kauers in 2007. To solve this question, it suffices to compute the asymptotics of the diagonal coefficients, which can be done by the multivariate singularity analysis developed by Baryshnikov, Pemantle and Wilson. Note that the ultimate positivity is a necessary condition for the positivity, and therefore can be used to either exclude the nonpositive cases or further support the conjectural positivity.Comment: 5 pages, extended abstrac

Some Rapidly Converging Series for ζ(2n+1)\zeta(2n+1) from Abstract Operators

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).$$ The method is based on the mapping relationships between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relationships between power series and trigonometric series, if each coefficient of a power series is respectively equal to that of a trigonometric series. Thus we obtain a general method to find the sum of the Dirichlet series of integer variables. By defining the Zeta function in an abstract operators form, we have further generalized these results on the whole complex plane.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1008.504

A note on Fox's H function in the light of Braaksma's results

In our previous works we found a power series expansion of a particular case of Fox's $H$ function $H^{q,0}_{p,q}$ in a neighborhood of its positive singularity. An inverse factorial series expansion of the integrand of $H^{q,0}_{p,q}$ served as our main tool. However, a necessary restriction on parameters is missing in those works. In this note we fill this gap and give a simpler and shorter proof of the expansion around the positive singular point. We further identify more precisely the abscissa of convergence of the underlying inverse factorial series. Our new proof hinges on a slight generalization of a particular case of Braaksma's theorem about analytic continuation of Fox's $H$ function.Comment: 12 pages; no figure

Perturbing PLA

We proved earlier that every measurable function on the circle, after a uniformly small perturbation, can be written as a power series (i.e. a series of exponentials with positive frequencies), which converges almost everywhere. Here we show that this result is basically sharp: the perturbation cannot be made smooth or even H\"older. We discuss also a similar problem for perturbations with lacunary spectrum.Comment: 21 page

Bilinear generating functions for orthogonal polynomials

Using realisations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner-Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner-Pollaczek polynomials. For the positive discrete series representations of the quantised universal enveloping algebra Uq(su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey-Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials.Comment: 18 pages, LaTe

Sums of quadratic functions with two discriminants

Zagier in [4] discusses a construction of a function $F_{k,D}(x)$ defined for an even integer $k \geq 2$, and a positive discriminant $D$. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the $D$-th Fourier coefficient of weight $k+1/2$ Eisenstein series constructed by H. Cohen in \cite{Cohen}. In this note we consider a construction which works both for even and odd positive integers $k$. Our function $F_{k,D,d}(x)$ depends on two discriminants $d$ and $D$ with signs sign$(d)=$ sign$(D)=(-1)^k$, degenerates to Zagier's function when $d=1$, namely, $$F_{k,D,1}(x)=F_{k,D}(x),$$ and has very similar properties. In particular, we prove that the average value of $F_{k,D,d}(x)$ is again a Fourier coefficient of H. Cohen's Eisenstein series of weight $k+1/2$, while now the integer $k \geq 2$ is allowed to be both even and odd