p-adic gamma function

In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita (1975), though Boyarsky (1980) pointed out that Dwork (1964) implicitly used the same function. Diamond (1977) defined a p-adic analog Gp of log Γ. Overholtzer (1952) had previously given a definition of a different p-adic analogue of the gamma function, but his function does not have satisfactory properties and is not used much.

DefinitionEdit

The p-adic gamma function is the unique continuous function of a p-adic integer x (with values in  ) such that

 

for positive integers x, where the product is restricted to integers i not divisible by p. As the positive integers are dense with respect to the p-adic topology in  ,   can be extended uniquely to the whole  . Here   is the ring of p-adic integers. It comes by the definition that the values of   are invertible in  . This is so, because these values are products of integers not divisible by p, and this property holds after the continuous extension to  . Thus  . Here   is the set of invertible p-adic integers.

Basic properties of Edit

The classical gamma function satisfies the functional equation   for any  . This has an analogue with respect to the Morita gamma function:

 

The Euler's reflection formula   has its following simple counterpart in the p-adic case:

 

where   is the first digit in the p-adic expansion of x, unless  , in which case   rather than 0.

Special valuesEdit

 
 
 
 

and, in general,

 

At   the Morita gamma function is related to the   Legendre symbol:

 

It can also be seen, that   hence   as  .[1]:369

Other interesting special values come from the Gross–Koblitz formula, which was first proved by cohomological tools, and later was proved using more elementary methods.[2] For example,

 
 

where   denotes the root with first digit 3, and with   we denote the root with first digit 2. (Such specifications must always be done if we talk about roots.)

Another example is

 

where   is the square root of   in   congruent to 1 modulo 3.[3]

p-adic Raabe formulaEdit

The Raabe-formula for the classical Gamma function says that

 

This has an analogue for the Iwasawa logarithm of the Morita gamma function:[4]

 

The ceiling function to be understood as the p-adic limit   such that   through rational integers.

Mahler expansionEdit

The Mahler expansion is similarly important for p-adic functions as the Taylor expansion in classical analysis. The Mahler expansion of the p-adic gamma function is the following:[1]:374

 

where the sequence   is defined by the following identity:

 

See alsoEdit

ReferencesEdit

  • Boyarsky, Maurizio (1980), "p-adic gamma functions and Dwork cohomology", Transactions of the American Mathematical Society, 257 (2): 359–369, doi:10.2307/1998301, ISSN 0002-9947, JSTOR 1998301, MR 0552263
  • Diamond, Jack (1977), "The p-adic log gamma function and p-adic Euler constants", Transactions of the American Mathematical Society, 233: 321–337, doi:10.2307/1997840, ISSN 0002-9947, JSTOR 1997840, MR 0498503
  • Diamond, Jack (1984), "p-adic gamma functions and their applications", in Chudnovsky, David V.; Chudnovsky, Gregory V.; Cohn, Henry; et al. (eds.), Number theory (New York, 1982), Lecture Notes in Math., 1052, Berlin, New York: Springer-Verlag, pp. 168–175, doi:10.1007/BFb0071542, ISBN 978-3-540-12909-7, MR 0750664
  • Dwork, Bernard (1964), "On the zeta function of a hypersurface. II", Annals of Mathematics, Second Series, 80 (2): 227–299, doi:10.2307/1970392, ISSN 0003-486X, JSTOR 1970392, MR 0188215
  • Morita, Yasuo (1975), "A p-adic analogue of the Γ-function", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 22 (2): 255–266, hdl:2261/6494, ISSN 0040-8980, MR 0424762
  • Overholtzer, Gordon (1952), "Sum functions in elementary p-adic analysis", American Journal of Mathematics, 74 (2): 332–346, doi:10.2307/2371998, ISSN 0002-9327, JSTOR 2371998, MR 0048493
  1. ^ a b Robert, Alain M. (2000). A course in p-adic analysis. New York: Springer-Verlag.
  2. ^ Robert, Alain M. (2001). "The Gross-Koblitz formula revisited". Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova. 105: 157–170. doi:10.1016/j.jnt.2009.08.005. hdl:2437/90539. ISSN 0041-8994. MR 1834987.
  3. ^ Cohen, H. (2007). Number Theory. 2. New York: Springer Science+Business Media. p. 406.
  4. ^ Cohen, Henry; Eduardo, Friedman (2008). "Raabe's formula for p-adic gamma and zeta functions". Annales de l'Institut Fourier. 88 (1): 363–376. doi:10.5802/aif.2353. MR 2401225.