Adic Gamma Research Articles

Some q-supercongruences related to Swisher’s (H.3) conjecture

We first give a [Formula: see text]-analogue of a supercongruence of Sun, which is a generalization of Van Hamme’s (H.2) supercongruence for any prime [Formula: see text]. We also give a further generalization of this [Formula: see text]-supercongruence, which may also be considered as a generalization of a [Formula: see text]-supercongruence recently conjectured by the second author and Zudilin. Then, by combining these two [Formula: see text]-supercongruences, we obtain [Formula: see text]-analogues of the following two results: for any integer [Formula: see text] and prime [Formula: see text] with [Formula: see text] [Formula: see text] [Formula: see text] which are generalizations of Swisher’s (H.3) conjecture modulo [Formula: see text] for [Formula: see text]. The key ingredients in our proof are the ‘creative microscoping’ method, the [Formula: see text]-Dixon sum, Watson’s terminating [Formula: see text] transformation, and properties of the [Formula: see text]-adic Gamma function.

Three pairs of congruences concerning sums of central binomial coefficients

Recently the first author proved a congruence proposed in 2006 by Adamchuk: [Formula: see text] for any prime [Formula: see text]. In this paper, we provide more examples (with proofs) of congruences of the same kind [Formula: see text] where [Formula: see text] is a prime such that [Formula: see text], [Formula: see text] is a fraction in [Formula: see text] and [Formula: see text] is a [Formula: see text]-adic integer. The key ingredients are the [Formula: see text]-adic Gamma function [Formula: see text] and a special class of computer-discovered hypergeometric identities.

A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-adic Gamma function.

On applications for Mahler expansion associated with p-adic q-integrals

In this paper, we primarily consider a generalization of the fermionic [Formula: see text]-adic [Formula: see text]-integral on [Formula: see text] including the parameters [Formula: see text] and [Formula: see text] and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of [Formula: see text]-Changhee polynomials and numbers as [Formula: see text]-Changhee polynomials and numbers with weight [Formula: see text] and [Formula: see text]-Changhee polynomials and numbers of second kind with weight [Formula: see text]. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted [Formula: see text]-Euler polynomials, [Formula: see text]-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted [Formula: see text]-Changhee polynomials and [Formula: see text]-adic gamma function. Also, we compute the weighted fermionic [Formula: see text]-adic [Formula: see text]-integral of the derivative of [Formula: see text]-adic gamma function. Moreover, we give a novel representation for the [Formula: see text]-adic Euler constant by means of the weighted [Formula: see text]-Changhee polynomials and numbers. We finally provide a quirky explicit formula for [Formula: see text]-adic Euler constant.

Supercongruences for truncated hypergeometric series and p-adic gamma function

AbstractWe prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

SUPERCONGRUENCES INVOLVING -ADIC GAMMA FUNCTIONS

We establish some supercongruences for the truncated $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series involving the $p$-adic gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated $_{3}F_{2}$ hypergeometric series. Related supercongruences modulo $p^{3}$ are proposed as conjectures.

A p-adic Supercongruence for Truncated Hypergeometric Series $${}_7F_6$$ 7 F 6

Using an identity due to Gessel and Stanton and some properties of the $p$-adic Gamma function, we establish a $p$-adic supercongruence for truncated hypergeometric series ${}_7F_6$. From it we deduce some related supercongruences, which extend certain recent results and confirm a supercongruence conjecture.

Analogues of the binomial coefficient theorems of Gauss and Jacobi

The theorems of Gauss and Jacobi that give modulo [Formula: see text] evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo [Formula: see text]. In this paper, we further extend these results to congruences modulo [Formula: see text]. In the process, we prove congruences to arbitrarily high powers of [Formula: see text] for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's [Formula: see text]-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

Supercongruences and truncated hypergeometric series

We employ the $p$-adic Gamma function and some formulas on hypergeometric series to establish several new supercongruences.

On the supercongruence conjectures of van Hamme

In 1997, van Hamme developed $$p$$ –adic analogs, for primes p, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the $$p$$ –adic gamma function, and are called Ramanujan-type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions.

Hypergeometric type identities in the $p$-adic setting and modular forms

We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.

The trace of Frobenius of elliptic curves and thep-adic gamma function

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p >3, that the trace of Frobenius of any elliptic curve over Fp, whose j -invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over Fp when p 1 .mod 12/. 1. Introduction and statement of results Let Fp denote the finite field with p, a prime, elements. Consider E=Q an elliptic curve with an integral model of discriminantA.E/. We denote Ep the reduction of E modulo p. We note that Ep is nonsingular, and hence an elliptic curve over Fp, if and only if p› A.E/, in which case we say p is a prime of good reduction. Regardless, we define

Some Properties of the -Extension of the -Adic Gamma Function

We study the -extension of the -adic gamma function . We give a new identity for the -extension of the -adic gamma in the case . Also, we derive some properties and new representations of the -extension of the -adic gamma in general case.

Frobenius map and the p -adic Gamma function

In this note we study the relationship between the power series expansion of the Dwork exponential and the Mahler expansion of the p -adic Gamma function. We exploit this relationship to prove that certain quantities that appeared in our computations in Shapiro (2009) [12] can be expressed in terms of the derivatives of the p -adic Gamma function at 0. We then use this to prove the claim made in Shapiro (2009) [12] about the non-trivial off-diagonal entry in the Frobenius matrix of the mirror quintic threefold.

The $p$-adic gamma measures

The p-adic log gamma function and its derivatives are used to define distributions and measures on the p-adic units. These measures are then used to interpolate the Leopoldt-Kubota p-adic L-functions on the positive integers.