The object of this chapter is to introduce the padic gamma function. During his lifetime, ramanujan provided many formulae relating binomial sums to special values of the gamma function. In section 3, we establish several basic properties of the p adic multiple zeta and log gamma functions, many of which are close analogues of the corresponding complexvalued functions. Some properties of the extension of the adic gamma function. We then prove several fundamental results for these padic log gamma functions, including the laurent series expansion, the distribution formula, the functional equation and the reflection formula. On the p adic gamma function and changhee polynomials. Rodriguezvillegas for making me interested in the whole subject thanks to a gp script for computing moritas padic gamma function, to f.
Based on numerical computations, van hamme recently conjectured padic analogues to such formulae. Moreover, we discover qvolkenborn integral of the derivative of p. Mortenson using the theory of finite field hypergeometric series follows from one of our more general. Greenberg, on the behavior of adic lfunctions at s 0, invent, math. Furthermore, we provide a limit representation of aforementioned euler. Gamma function the eulerian integral,n0 is called gamma function and is denoted by example. Our treatment is strongly influenced by boyarsky 5. Log gamma function, euler constants, padic functions.
It showcases research results in functional analysis over nonarchimedean valued complete fields. In section 3, we establish several basic properties of the padic multiple zeta and log gamma functions, many of which are close analogues of the corresponding complexvalued functions. A note on the padic gamma function and qchanghee polynomials. In the present work we consider a p adic analogue of the classical beta function by using y. On a relation between padic gamma functions and padic periods. We prove three more general supercongruences between truncated hypergeometric series and padic gamma function from which some known supercongruences follow. The gamma and the beta function as mentioned in the book 1, see page 6, the integral representation 1. Section 0 covers the background material on drinfeld modules. Pdf the boole polynomials associated with the padic. The adic gamma function had been studied by diamond, barsky, and others. A supercongruence conjectured by rodriguezvillegas and proved by e. Moreover, we give relationship between p adic qlog gamma funtions and qextension of genocchi numbers and polynomials. In the present work we consider a padic analogue of the classical beta function by using y.
The gamma and the beta function delft university of. We investigate several properties and relationships belonging to the foregoing gamma function. Pdf a note on the qanalogue of kims padic log gamma. Supercongruences for truncated hypergeometric series and p. Moreover, we give relationship between padic qlog gamma funtions and qextension of genocchi numbers and polynomials. Connections with padic lfunctions and expansions of eisenstein series are discussed. I gave a version of dworks first paper on the zeta function of a nonsingular. Colmez for his help on proving the results of section 11.
Diamond, the 7adic log gamma function and 7adic euler constants, trans. We also express various formalgroup congruences, including those involving apery numbers, in. Finally, using the padic diamondeuler log gamma functions, we obtain the formula for the derivative of the padic hurwitztype euler zeta function at, then we show that the padic hurwitz. The extension of the adic gamma function is defined by koblitz as follows. Some special properties of the gamma function are the following.
On a qanalogue of the padic log gamma functions and related. A study on novel extensions for the padic gamma and padic. These notes are essentially the lecture notes for that course. By wellknown methods such formulae yield expressions for roots of congruence zetafunctions in terms of jacobi sums in the nonsingular case. According to godefroy 9, eulers constant plays in the gamma function theory a similar role as. On padic gamma function related to qdaehee polynomials. 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. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. In this note we study the relationship between the. Finally, using the padic diamondeuler log gamma functions, we obtain the formula for the derivative of the padic hurwitztype euler zeta function at s 0, then we show that the padic hurwitztype euler zeta functions will appear in the studying for a special case of padic analogue of the s, tversion of the abelian rank one stark. Transcendental values of the padic digamma function. In this paper, we introduce qanalogue of p adic log gamma func tions with weight alpha.
The first term, i presented several classical results on zeta functions in characteristic p weils calculation of the zeta. In mathematics, a padic zeta function, or more generally a padic lfunction, is a function analogous to the riemann zeta function, or more general lfunctions, but whose domain and target are padic where p is a prime number. On a relation between padic gamma functions and padic. We will use q, qp, z, zp, c and clp for, respectively, the field of rational numbers, the padic completion of q, the ring of rational integers, the padic completion of z in qp, the field of. The padic series are compared with the analogous classical expansions. Here log p denotes iwasawas padic log function and. In this paper, we introduce qanalogue of padic log gamma func tions with weight alpha. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Now we describe more precisely the contents of this paper. The main tool is the padic maximum principle treated in the. Readers may refer to gk or artins book on the classical gamma function. We shall study gextensions of the function derivative of loggamma and its twists by roots of unity, dirichlet characters, etc. We also derived two qvolkenborn integrals of padic gamma function in terms of qdaehee polynomials and numbers and qdaehee polynomials and numbers of the second kind.
A presentation of results in padic banach spaces, spaces over fields with an infinite rank valuation, frechet and locally convex spaces with schauder bases, function spaces, padic harmonic analysis, and related areas. Dasguptadarmonpollack ddp proved that conjecture 1. Furthermore the relationship between the padic gamma function and changhee polynomials and also between the changhee polynomials and padic euler constants is obtained. In this paper, using the fermionic padic integral on z p, we define the corresponding padic log gamma functions, socalled padic diamondeuler log gamma functions. On a qanalogue of the padic log gamma functions and. Pdf the main aim of this paper is to set some correlations between boole polynomials and p adic gamma function in conjunction with p adic euler. This will be used to explain the padic beta function and ultimately for our discussion of singular disks in chapters 2426. Growth order, type and cotype of padic entire functions.
Pdf in this paper, we investigate several relations for padic gamma function by means of their mahler expansion and fermionic padic. We shall study gextensions of the function derivative of log gamma and its twists by roots of unity, dirichlet characters, etc. A presentation of results in p adic banach spaces, spaces over fields with an infinite rank valuation, frechet and locally convex spaces with schauder bases, function spaces, p adic harmonic analysis, and related areas. Gamma function the factorial function can be extended to include noninteger arguments through the use of eulers second integral given as z. Some of the historical background is due to godefroys beautiful essay on this function 9 and the more modern textbook 3 is a complete study. The relationship between the padic gamma function and qchanghee numbers is obtained. On padic multiple zeta and log gamma functions request pdf. Pdf a note on the padic gamma function and qchanghee. It was first explicitly defined by morita 1975, though boyarsky 1980 pointed out that dwork 1964 implicitly used the same function. We will be brief since similar material has been discussed by lang 23 and by katz 21. 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. Research article some properties of the extension of the. In 7, koblitz constructed a qanalogue of the p adic l function l p, q s, and suggested two questions.
A padic lfunction arising in this way is typically called an arithmetic padic lfunction as it encodes arithmetic data of the galois module involved. The trace of frobenius of elliptic curves padic gamma function. A padic analogue of a formula of ramanujan dermot mccarthy and robert osburn abstract. For example, the domain could be the padic integers z p, a profinite pgroup, or a padic family of galois representations, and the image. We prove three more general supercongruences between truncated hypergeometric series and p adic gamma function from which some known supercongruences follow. We will use q, qp, z, zp, c and clp for, respectively, the field of rational numbers, the p adic completion of q, the ring of rational integers, the p adic completion of z in qp, the field of. The main conjecture of iwasawa theory now a theorem due to barry mazur and andrew wiles is the statement that the kubotaleopoldt p adic l function and an arithmetic analogue constructed. Also, we derive some properties and representations for the extension of the adic gamma function. Pdf on mahler expansion of padic gamma function affiliated. A study on novel extensions for the padic gamma and p. In addition, the padic euler constants are expressed in term of mahler coefficients of the p. Pdf in the present work, we consider the fermionic padic qintegral of padic gamma function and the derivative of padic gamma function by using. On a relation between padic gamma functions and padic periods tomokazu kashio, tokyo univ.
In this paper, we prove that the qanalogue of bernoulli numbers occur in the coefficients of some stirling type series for p adic analytic qlog gamma functions. We also define p adic euler constants and use them to obtain results on g p and on the logarithmic derivative of moritas tp. On padic gamma function related to qdaehee polynomials and. Transcendental values of the padic digamma function by m. 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. Friedman for very interesting discussions, and especially to p. Connections with p adic l functions and expansions of eisenstein series are discussed. We obtain some elementary properties of the p adic beta function. 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. Saradha mumbai in honour of professor wolfgang schmidt 1. Gamma functions for function fields and drinfeld modules. Diamond 1977 defined a p adic analog g p of log overholtzer 1952 had previously given a definition of a different p adic analogue of the.
Notes on padic lfunctions 3 with a continuous, e linear g qaction. The p adic series are compared with the analogous classical expansions. One notable exception is the re ection functional equation theorem 3. Connections with p adic lfunctions and expansions of eisenstein series are discussed. Its possible to show that weierstrass form is also valid for complex numbers. This is the logarithmic derivative of eulers function. On padic diamondeuler log gamma functions sciencedirect. Question 1 was solved by satoh 8, but question 2 still remains open. Relation between gamma and factorial other results. Connections with padic lfunctions and expansions of eisenstein series are. The relationship between some special functions and the adic gamma function were investigated by gross and koblitz, cohen and friedman. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p 2.
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