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