P-adic Gamma Function Research Articles

On the differential transcendentality of the Morita p-adic gamma function

In p-adic analysis one can find an analog of the classical gamma function defined on the ring of p-adic integers. In 1975, Morita defined the p-adic gamma function Gamma _p by a suitable modification of the function n mapsto n!. In this note we prove that for any given prime number p the Morita p-adic gamma function Gamma _p is differentially transcendental over {mathbb {C}}_p(X). The main result is an analog of the classical Hölder’s theorem, which states that Euler’s gamma function Gamma does not satisfy any algebraic differential equation whose coefficients are rational functions.

Zeros of hypergeometric functions in the p-adic setting

McCarthy (Pac J Math 261(1):219–236, 2013) defined hypergeometric functions in the p-adic setting over finite fields using p-adic gamma functions. These functions possess many properties that are analogous to classical hypergeometric type identities. In this paper, we investigate values of two generic families of these hypergeometric functions that we denote by {_nG_n}(t)_p and {_n{widetilde{G}}_n}(t)_p for nge 3, and tin {mathbb {F}}_p, the finite field with p elements. These results generalize special cases of p-adic analogues of Whipple’s theorem and Dixon’s theorem of classical hypergeometric series. We also examine zeros of the functions {_nG_n}(t)_p, and {_n{widetilde{G}}_n}(t)_p over {mathbb {F}}_p. Moreover, we classify the values of t for which {_nG_n}(t)_p=0 for infinitely many primes. For example, we show that there are infinitely many primes for which {_{2k}G_{2k}}(-1)_p=0. In contrast, for tne 0 there are no primes for which {_{2k}{widetilde{G}}_{2k}}(t)_p=0.

L-functions of twisted exponential sums over finite fields

Let $${\mathbb {F}}_{q}$$ be the finite field of q elements and $$\chi _{1},\ldots ,\chi _{n}$$ the multiplicative characters of $${\mathbb {F}}_{q}$$ . Given a Laurent polynomial $$f(X)\in {\mathbb {F}}_q[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$$ , the corresponding L-function is defined to be $$\begin{aligned} L^{*}(\chi _{1},\ldots ,\chi _{n},f;T) =\exp \Big (\sum \nolimits _{h=1}^{\infty }S^{*}_{h}(\chi _{1},\ldots ,\chi _{n},f)\frac{T^{h}}{h}\Big ), \end{aligned}$$ where $$S^{*}_{h}(\chi _{1},\ldots ,\chi _{n},f)$$ is the twisted exponential sum defined in the extension of $${\mathbb {F}}_{q}$$ of degree h. In this paper, we obtain the explicit formulae for $$L^{*}(\chi _{1},\ldots ,\chi _{n},f;T)$$ for the Laurent polynomials with full column rank exponent matrix in terms of p-adic gamma functions, which generalizes the results of Wan, Hong and Cao. We also evaluate the slopes of the reciprocal zeros and reciprocal poles of $$L^{*}(\chi _{1},\ldots ,\chi _{n},f;T)$$ and determine the p-adic Newton polygons of the polynomials associated to the L-function $$L^{*}(\chi _{1},\ldots ,\chi _{n},f;T)$$ .

On p-adic gamma function related to q-Daehee polynomials and numbers

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on of p-adic gamma function via their Mahler expansions. We also derived two -Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p -adic Euler constant by means of the q-Daehee polynomials and numbers.

Congruences for the truncated Appell series F3 and F4

In this paper, we establish two congruences on the truncated Appell series and , one of which relates the truncated Appell series to the values of p-adic Gamma functions.

A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions

In this paper, we introduce the ρ , q -analog of the p-adic factorial function. By utilizing some properties of ρ , q -numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic ρ , q -gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2 . We also derive more representations of the p-adic ρ , q -gamma function in general case. Moreover, we consider the p-adic ρ , q -Euler constant derived from the derivation of p-adic ρ , q -gamma function at x = 1 . Furthermore, we provide a limit representation of aforementioned Euler constant based on ρ , q -numbers. Finally, we consider ρ , q -extension of the p-adic beta function via the p-adic ρ , q -gamma function and we then investigate various formulas and identities.

The Boole polynomials associated with the p-adic gamma function

We set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on Zp. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.

On the <i>p</i>-adic Gamma Function and Changhee Polynomials

The p-adic gamma function is considered to obtain its derivative and to evaluate its the fermionic p-adic integral. Furthermore the relationship between the p-adic gamma function and Changhee polynomials and also between the Changhee polynomials and p-adic Euler constants is obtained. In addition, the p-adic Euler constants are expressed in term of Mahler coefficients of the p-adic gamma function.

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.

A note on the p-adic gamma function and q-Changhee polynomials

A note on the p-adic gamma function and q-Changhee polynomials

Summation identities and transformations for hypergeometric series

We find summation identities and transformations for the McCarthy’s p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$\begin{aligned} Z_{\lambda }: x_1^d+x_2^d=d\lambda x_1x_2^{d-1} \end{aligned}$$ over a finite field $$\mathbb {F}_p$$ . Salerno expresses the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of quotients of p-adic gamma functions under the condition that $$d|p-1$$ . In this paper, we first express the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of McCarthy’s p-adic hypergeometric series for any odd prime p not dividing $$d(d-1)$$ , and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene’s finite field hypergeometric series.

Congruences concerning truncated hypergeometric series

We employ some formulae on hypergeometric series and p-adic Gamma function to establish several congruences.

Supercongruences on Truncated Hypergeometric Series

In this paper, we use p-adic Gamma function and certain formulas on hypergeometric series to establish several new supercongruences. In particular, we give a generalization of a p-adic supercongruence conjecture due to van Hamme and Swisher.

Supercongruences arising from basic hypergeometric series

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

Infinitesimal deformation of p-adic differential equations on Berkovich curves

We show that if a differential equation $$\mathscr {F}$$ over a quasi-smooth Berkovich curve X has a certain compatibility condition with respect to an automorphism $$\sigma $$ of X, then $$\mathscr {F}$$ acquires a semi-linear action of $$\sigma $$ (i.e. lifting that on X). The compatibility condition forces the automorphism $$\sigma $$ to be close to the identity of X, so the above construction applies to a certain class of automorphisms called infinitesimal. This generalizes Andre and Di Vizio (Asterisque 1(296):55–111, 2004) and Pulita (Compos. Math. 144(4):867–919, 2008). We also obtain an application to Morita’s p-adic Gamma function, and to related values of p-adic L-functions.

Some supercongruences occurring in truncated hypergeometric series

For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of p-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae, most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of Γ-values. At this point sums have become quotients. Writing these Γ-quotients as Γp-quotients, we are in a situation that is well-suited for proving p-adic congruences. These Γp-functions can be p-adically approximated by their Taylor series expansions. Sometimes there is cancellation of the lower order terms, leading to stronger congruences. Using this technique we prove, among other things, a conjecture of Kibelbek and a strengthened version of a conjecture of van Hamme.

Some congruences on truncated hypergeometric series

In this paper, we use p-adic Gamma function and certain formulas on hypergeometric series to establish several new supercongruences. In particular, we give a generalization of a p-adic supercongruence conjecture due to van Hamme and Swisher.

Summation identities and special values of hypergeometric series in the p-adic setting

We prove hypergeometric type summation identities for a function defined in terms of quotients of the p-adic gamma function by counting points on certain families of hyperelliptic curves over Fq. We also find certain special values of that function.

Certain transformations for hypergeometric series in the p-adic setting

In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

P-adic Gamma function and the polynomials [formula omitted] and [formula omitted] over [formula omitted

In [14], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. He expressed the trace of Frobenius of elliptic curves in terms of special values of G22[⋯]. For d⩾2, we establish four different expressions for the number of distinct zeros of the polynomials xd+ax+b and xd+axd−1+b over Fq in terms of special values of the function Gd−1d−1[⋯].