Some Properties of the -Extension of the -Adic Gamma Function

Abstract

We study the -extension of the -adic gamma function . We give a new identity for the -extension of the -adic gamma in the case . Also, we derive some properties and new representations of the -extension of the -adic gamma in general case.

1. Introduction

Let be a prime number and let , and denote the ring of -adic integers, the field of -adic numbers, and the completion of the algebraic closure of , respectively. It is well known that the analogous of the classical gamma function in -adic context depends on modifying the factorial function [1]. The factorial function in is defined as The -adic gamma function is defined by Morita [2] as the continuous extension to of the function . That is, is defined by the formula for , where approaches through positive integers. The -adic gamma function had been studied by Diamond [3], Barsky [4], and others. The relationship between some special functions and the -adic gamma function were investigated by Gross and Koblitz [5], Cohen and Friedman [6]. and Shapiro [7].

The -extension of the -adic gamma function is defined by Koblitz as follows.

Definition 1 (see [8]). Let . The -extension of the -adic gamma function is defined by formula for , where approaches through positive integers. We recall that .

The -extension of the -adic gamma function was studied by Koblitz [8, 9], Nakazato [10], Kim et al. [11], and Kim [12].

2. Main Results

In the present work, we give a new identity for the -extension of the -adic gamma function in special case . Also, we derive some properties and representations for the -extension of the -adic gamma function .

Theorem 2. If , then for all where is defined by the formula

Proof. Let and . From Proposition  3 in [12] we known that Here, is the greatest integer function. Taking in place of , the relation becomes Now, let in base 2. If , then in base 2 and Thus, we get If , then Hence, Thus, we have and thus, we obtain

We recall that the -factorial is defined in [13] by the formula for , where Note that for , we can define .

We use the following theorem to prove our results.

Theorem 3 (see [12]). Let . Then, where is the greatest integer function. In particular,

Theorem 4. Let and let be the sum of the digits of in base . Then (a)(b) .

Proof. From the Theorem 3 we know that By taking instead of , respectively, we get the relations By multiplying of the equalities above, we can easily obtain Therefore, we get the relation (a) Therefore, we get the relation (b)

Theorem 5. Let and let . Then

Proof. From Theorem 3 it follows that Taking of instead of , respectively, we have the equalities By multiplying of the equalities above, we can easily obtain Thus,

Lemma 6. Let , , and let be a prime number. Then, for

Proof. For For it follows that
Then, we obtain

Theorem 7. Let and let be the sum of the digits of in base . Then

Proof. This theorem can be proved by using Theorem 4 and Lemma 6.

Acknowledgments

This work is supported by Mersin University and the Scientific and Technological Research Council of Turkey (TÜBİTAK). The authors would like to thank the reviewers for their useful comments and suggestions.

References

1. W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic Analysis, vol. 4, Cambridge University Press, Cambridge, UK, 1984. View at: Zentralblatt MATH | MathSciNet
2. Y. Morita, “A $p$-adic analogue of the $\mathrm{\Gamma }$-function,” Journal of the Faculty of Science, vol. 22, no. 2, pp. 255–266, 1975. View at: Google Scholar | MathSciNet
3. J. Diamond, “The $p$-adic log gamma function and $p$-adic Euler constants,” Transactions of the American Mathematical Society, vol. 233, pp. 321–337, 1977. View at: Google Scholar | MathSciNet
4. D. Barsky, “On Morita's $p$-adic gamma function,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 89, no. 1, pp. 23–27, 1981. View at: Publisher Site | Google Scholar | MathSciNet
5. B. H. Gross and N. Koblitz, “Gauss sums and the $p$-adic $\mathrm{\Gamma }$-function,” Annals of Mathematics, vol. 109, no. 3, pp. 569–581, 1979. View at: Publisher Site | Google Scholar | MathSciNet
6. H. Cohen and E. Friedman, “Raabe's formula for $p$-adic gamma and zeta functions,” Annales de l'Institut Fourier, vol. 58, no. 1, pp. 363–376, 2008. View at: Publisher Site | Google Scholar | MathSciNet
7. I. Shapiro, “Frobenius map and the $p$-adic gamma function,” Journal of Number Theory, vol. 132, no. 8, pp. 1770–1779, 2012. View at: Publisher Site | Google Scholar | MathSciNet
8. N. Koblitz, “q-extension of the p-adic gamma function,” Transactions of the American Mathematical Society, vol. 260, no. 2, 1980. View at: Google Scholar
9. N. Koblitz, “$q$-extension of the $p$-adic gamma function. II,” Transactions of the American Mathematical Society, vol. 273, no. 1, pp. 111–129, 1982. View at: Publisher Site | Google Scholar | MathSciNet
10. H. Nakazato, “The $q$-analogue of the $p$-adic gamma function,” Kodai Mathematical Journal, vol. 11, no. 1, pp. 141–153, 1988. View at: Publisher Site | Google Scholar | MathSciNet
11. T. K. Kim et al., “A note on analogue of gamma functions,” in Proceedings of the 5th Transcendental Number Theory, pp. 111–118, Gakushin University, Tokyo, Japan, 1997. View at: Google Scholar
12. Y. S. Kim, “$q$-analogues of $p$-adic gamma functions and $p$-adic Euler constants,” Korean Mathematical Society, vol. 13, no. 4, pp. 735–741, 1998. View at: Google Scholar | MathSciNet
13. R. Askey, “The $q$-gamma and $q$-beta functions,” Applicable Analysis, vol. 8, no. 2, pp. 125–141, 1978/79. View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2013 Hamza Menken and Adviye Körükçü. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.