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Single variable hypergeometric functions _pF_q arise in connection with the power series solution of the Schrödinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to obtain analytic expressions, and we present the reduction of a number of cases of _pF_p and _p+1F_p, mainly for p=2 and p=3. These and related series have additional applications in quantum and statistical physics and chemistry.

We study the spectral zeta functions associated to the radial Schrödinger problem with potential V(x) = x^2M + αx^M-1 + (λ² - 1/4)/x². After directly computing some of the zeta functions, we use the quantum Wronskian equation to give sum rules between them, allowing for instances where the explicit form of the zeta functions can be simplified. An immediate application of this work is to derive functional relations and identities involving hypergeometric series, allowing for known identities to be found as instances of more general results. Our work is then extended to a class of related -symmetric eigenvalue problems. Using the fused quantum Wronskian, we give a simple method for indirectly calculating the associated spectral zeta functions. This method is then applied to calculate the nonlocal integrals of motion G_n which appear in an associated integrable quantum field theory.

This paper investigates a new special function referred to as the error zeta function. Derived as a fractional generalization of hypergeometric zeta functions, the error zeta function is shown to exhibit many properties analogous to its hypergeometric counterpart, including its intimate connection to Bernoulli numbers. These new properties are treated in detail and used to demonstrate a pre-functional equation satisfied by this special function.

By means of two hypergeometric summation formulae, we establish two large classes of infinite series identities with harmonic numbers and central binomial coefficients. Up to now, these numerous formulae have hidden behind very few known identities of Apéry-like series for Riemann-zeta function, discovered mainly by Lehmer [14] and Elsner [12] as well as Borwein et al. [4, 5, 7]. All the computation and verification are carried out by an appropriately-devised Mathematica package.

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

Duality relations are deduced for tails of multiple-zeta values using elementary methods. These formulas extend the classical duality formulas for multiple-zeta values.

Kaneko and Zagier proved the relation between the Atkin orthogonal polynomials and supersingular j-polynomials for PSL₂(ℤ). Tsutsumi also proved its relation for the Hecke subgroups of level less than or equal to 4. In this paper, we define the Atkin inner product for the Fricke groups of level less than or equal to 3 and construct the Atkin orthogonal polynomials. Then, we give the relation between supersingular -polynomials defined by Koike and its polynomials. We also give extremal quasimodular forms of depth 1 by using its orthogonal polynomials.

Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan-type series for 1/π.

We provide various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

By applying the derivative operator to Whipple-type ${}_3{F}_2$-series identities and utilizing L’Hôspital rule, two families of summation formulae involving generalized harmonic numbers are established.

Using a modular equation of level $3$ and degree $23$ due to Chan and Liaw, we prove the fastest known (conjectured to be the fastest one) convergent rational Ramanujan-type series for $1/\pi$ of level $3$.

We define four functions $F_1^{\ast },F_2^{\ast },F_3^{\ast }$ and $F_4^{\ast }$ as finite field analogues of Appell series $F_1,F_2,F_3$ and $F_4$, respectively using purely Gauss sums in the spirit of finite field hypergeometric series introduced by McCarthy. We establish relations among $F_1^{\ast },F_2^{\ast }$ and $F_3^{\ast }$ analogous to those satisfied by the classical Appell series. Recently, several people have defined finite field analogues of Appell series using integral representations of Appell series. We show that our functions $F_1^{\ast },F_2^{\ast }$ and $F_3^{\ast }$ are closely related to those functions.

The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from $q=e^{2\pi i\tau }$ to the reciprocal of the elliptic modular $j$-function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by this change of parameters. Furthermore, by transforming these hypergeometric series expressions into a certain manageable form related to the Atkin(-like) polynomials and using the lemmas that appeared in the study of $p$-adic hypergeometric series by Dwork and Zudilin, the integrality problem can be reduced to the fact that a polynomial vanishes modulo a prime power, which we prove. We also prove that all extremal quasimodular forms of depth 1 with appropriate weight-dependent leading coefficients have integral Fourier coefficients by focusing on the hypergeometric expression of them.

Let $D_{\lambda }^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field ${𝔽}_q$ given by

Let p be an odd prime and $q=p^r$, $r\ge 1$. For positive integers n, let ${}_n{G}_n{[\cdots ]}_q$ denote McCarthy’s p-adic hypergeometric function. In this paper, we prove an identity expressing a ${}_4{G}_4{[\cdots ]}_q$ hypergeometric function as a sum of two ${}_2{G}_2{[\cdots ]}_q$ hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transformation that relates ${}_{n+2}{G}_{n+2}{[\cdots ]}_q$ and ${}_n{G}_n{[\cdots ]}_q$ hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of ${}_4{G}_4{[\cdots ]}_q$ and ${}_6{G}_6{[\cdots ]}_q$ hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.

Using a modular equation of level 3 and degree 23 due to Chan and Liaw, we prove the fastest known (conjectured to be the fastest one) convergent rational Ramanujan-type series for 1/π of level 3.