Rubin has a computation outside of the critical points for a cm elliptic curve in section 3. Table 2 contains the gamma function for arguments between 1 and 1. We also define padic euler constants and use them to obtain results on. Why exponential function on p adic numbers is meaningless.
Multiple gamma functions and derivatives of lfunctions at. Pdf on sep 1, 2015, hamza menken and others published gauss legendre multiplication. On a relation between padic gamma functions and padic. Request pdf a note on the padic loggamma functions in this paper we prove that qeuler numbers are occured in the coefficients of some stirling type seies for padic analytic qlog gamma. New padic hypergeometric functions and syntomic regulators. A study on novel extensions for the padic gamma and padic. Its importance is largely due to its relation to exponential and normal distributions. This is the logarithmic derivative of eulers function. On a relation between padic gamma functions and padic periods tomokazu kashio, tokyo univ.
Transcendental values of the padic digamma function 3 of l1,f and l p1,f see proposition 5. We define padic multiple zeta and log gamma functions using multiple volkenborn integrals, and develop some of their properties. On padic diamondeuler log gamma functions sciencedirect. Teichmuller character of x teichmullerx newton polygon of ffor prime p newtonpolyf, p parigp reference card parigp version 2. Euler 1729 as a natural extension of the factorial operation from positive integers to real and even complex values of this argument. Eulers product formula for the gamma function, combined with the functional equation and an identity for the eulermascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers abramowitz and stegun 6. Notes on padic lfunctions 3 with a continuous, e linear g qaction g q. Dirichlet euler product bterms direuler p a,b,expr p adic functions square of x, good for 2adics sqrx. The padic gamma function is considered to obtain its derivative and to evaluate its the fermionic padic integral. We also define p adic euler constants and use them to obtain results on g p and on the logarithmic derivative of. The p adic series are compared with the analogous classical expansions.
As in the complex case, it has an inverse function, named the p adic logarithm. Pdf a note on the qanalogue of kims padic log gamma type. Some properties of the q extension of the p adic gamma function menken, hamza and korukcu, adviye, abstract and applied analysis, 20. Moreover, we give relationship between padic qlog gamma funtions and q. Although our functions are close analogues of classical barnes multiple zeta and log gamma functions and have many properties similar to them, we nd that our p adic analogues also satisfy re.
Interpretation of thejadic log gamma function and euler. Furthermore the relationship between the padic gamma function and changhee polynomials and also between the changhee polynomials and padic euler constants is obtained. The p adic gamma function is considered to obtain its derivative and to evaluate its the fermionic p adic integral. The output is a \p\adic integer from dworks expansion, used to compute the \p\adic gamma function as in section 6. The padic analogue of the classical gamma function. The relationship between some special functions and the adic gamma function were investigated by gross and koblitz, cohen and friedman. Supercongruences for truncated hypergeometric series and p.
The coefficients of the expansion are now cached to speed up multiple evaluation, as in the trace formula for hypergeometric motives. The classical gamma function is firstly introduced by leonard euler 17071783 as r x. We define gp, a jadic analog of the classical log gamma function and show it satisfies relations similar to the standard formulas for log gamma. In addition, the padic euler constants are expressed in term of mahler coefficients of the padic gamma function. We express the derivative of padic euler lfunctions at s 0 and the special values of padic euler lfunctions at positive integers as linear combinations of padic diamondeuler log gamma functions. 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. The object of this chapter is to introduce the padic gamma function. Summation identities and transformations for hypergeometric series ii barman, rupam and saikia, neelam, functiones et approximatio commentarii mathematici, 2019. Pdf some properties of the padic beta function researchgate. As in the complex case, it has an inverse function, named the padic logarithm. Dec 06, 2008 the second author defined the absolute cmperiod symbol using the multiple gamma function. According to godefroy 9, eulers constant plays in the gamma function theory a similar role as. Moreover, we give relationship between p adic q log gamma funtions with weight alpha and beta and qextension of genocchi numbers. Let z, zp, tip denote, respectively, the adic integers, the p adic integers not divisible hyp, and the completion of the algebraic closure of the field of adic numbers.
We investigate several properties and relationships belonging to the foregoing gamma function. Using these functions, we prove the existence of a padic interpolation function that interpolates the qgeneralized polynomials. Thepadic log gamma function and padic euler constants1. Our treatment is strongly influenced by boyarsky 5. On a qanalogue of the padic log gamma functions and.
We define gp, a j adic analog of the classical log gamma function and show it satisfies relations similar to the standard formulas for log gamma. In mathematics, particularly p adic analysis, the p adic exponential function is a p adic analogue of the usual exponential function on the complex numbers. Moreover, we give relationship between p adic q log gamma funtions and qextension of genocchi numbers and polynomials. I think ive seen other cases, but generally it takes a lot of effort, i. Transcendental values of the padic digamma function by m. On a qanalogue of the padic log gamma functions and related. One more argument can be found in section 4 below, where a polynomial p e r is replaced by any entire function of order less than. Thepadic log gamma function and padic euler constants1 by jack diamond abstract. The classical raabe formula computes a definite integral of the logarithm of eulers.
The complex analytic polygamma functions are the rth derivative. The gamma distribution is another widely used distribution. Eulers gamma function the gamma function plays an important role in the functional equation for s that we will derive in the next chapter. Raabes formula for padic gamma and zeta functions numdam. In addition, the p adic euler constants are expressed in term of mahler coefficients of the p adic gamma function. Consequently, we define a padic twisted qlfunction which is a solution of a question of kim et al. We prove three more general supercongruences between truncated hypergeometric series and p adic gamma function from which some known supercongruences follow. Diamond 1977 defined a padic analog gp of log overholtzer 1952 had. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. The second author defined the absolute cmperiod symbol using the multiple gamma function. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. Connections with padic lfunctions and expansions of eisenstein series are discussed. Thep adic log gamma function and padic euler constants1 by jack diamond abstract.
It agrees with each of these on large parts of its domain and has the advantage of being a locally analytic function. The gamma function constitutes an essential extension of the idea of a factorial, since the argument z is not restricted to positive integer values, but can vary continuously. A note on the padic loggamma functions request pdf. We shall study gextensions of the function derivative of loggamma and its twists. In this paper, we introduce qanalogue of padic log gamma func tions with weight alpha. The fundamental aim of this paper is to describe qanalogue of p adic log gamma functions with weight alpha and beta. A new q analogue of bernoulli polynomials associated with p adic q integrals jang, leechae, abstract and applied analysis, 2008. The gamma function evalated at 1 2 is 1 2 p 4 the recursive relationship in 2 can be used to compute the value of the gamma function of all real numbers except the nonpositive integers by knowing only the value of the gamma function between 1 and 2. Its possible to show that weierstrass form is also valid for complex numbers. Request pdf on p adic multiple zeta and log gamma functions we define p adic multiple zeta and log gamma functions using multiple volkenborn integrals, and develop some of their properties. On the p adic gamma function and changhee polynomials. In the present chapter we have collected some properties of the gamma function. We then prove several fundamental results for these p adic log gamma functions, including the laurent series expansion, the distribution formula, the functional equation and the reflection formula.
Pdf gauss legendre multiplication formula for padic beta. This will be used to explain the padic beta function and ultimately for our discussion of singular disks in chapters 2426. Connections with p adic lfunctions and expansions of eisenstein series are discussed. Thep adic log gamma function and p adic euler constants1 by jack diamond abstract. Extension to of padic and l functions and padic measures on. Chapter 8 eulers gamma function universiteit leiden. Some properties of the extension of the adic gamma function. In this paper, we introduce qanalogue of p adic log gamma func tions with weight alpha. On padic multiple zeta and log gamma functions sciencedirect. Our idea is to bring forth the utility of this result along with several criteria that we gave in 9 for the nonvanishing of l1,f. The padic series are compared with the analogous classical expansions.
Request pdf a note on the padic loggamma functions in this paper we prove that qeuler numbers are occured in the coefficients of some stirling type seies for p adic analytic q log gamma. It was first explicitly defined by morita 1975, though boyarsky 1980 pointed out that dwork 1964 implicitly used the same function. The gamma function is applied in exact sciences almost as often as the well. The adic gamma function had been studied by diamond, barsky, and others. Transcendental values of the padic digamma function. Why exponential function on padic numbers is meaningless. In this paper, using the fermionic p adic integral on z p, we define the corresponding p adic log gamma functions, socalled p adic diamondeuler log gamma functions. Let \ p be the absolute value on up with \ p \ p p 1j. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except the nonpositive integers, where the function has simple poles. Z p, wherelog p ontherightdenotestheiwasawapadiclogarithmso log p p 0. Dx is not the logarithm of any function, but our notation is meant to recall the analogy with the classical log. On padic multiple zeta and log gamma functions request pdf. A distribution formula for kashios padic loggamma function. We recall the merita padic gamma function, r, defined on z by the initial.
We prove three more general supercongruences between truncated hypergeometric series and padic gamma function from which some known supercongruences follow. By the same motivation, we introduce q analogue of p adic log gamma type function with weight and we. A supercongruence conjectured by rodriguezvillegas and proved by e. This symbol is conjectured to give the same value as shimuras period symbol up to multiplication by algebraic numbers. Request pdf on padic multiple zeta and log gamma functions we define padic multiple zeta and log gamma functions using multiple volkenborn integrals, and develop some of their properties. As for the values of the padic lfunction at noncritical integers, thats much more mysterious. Saradha mumbai in honour of professor wolfgang schmidt 1. The extension of the adic gamma function is defined by koblitz as follows. A study on novel extensions for the padic gamma and p. Pdf in the present work we consider a padic analogue of the classical beta. Finally, using the padic diamondeuler log gamma functions, we obtain the formula for the derivative of the padic hurwitztype euler zeta. In mathematics, particularly padic analysis, the padic exponential function is a padic analogue of the usual exponential function on the complex numbers.