Research Article | Open Access
Some Properties of the -Extension of the -Adic Gamma Function
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.
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 . The factorial function in is defined as The -adic gamma function is defined by Morita  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 , Barsky , and others. The relationship between some special functions and the -adic gamma function were investigated by Gross and Koblitz , Cohen and Friedman . and Shapiro .
The -extension of the -adic gamma function is defined by Koblitz as follows.
Definition 1 (see ). Let . The -extension of the -adic gamma function is defined by formula for , where approaches through positive integers. We recall that .
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  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  by the formula for , where Note that for , we can define .
We use the following theorem to prove our results.
Theorem 3 (see ). 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
For it follows that
Then, we obtain
Theorem 7. Let and let be the sum of the digits of in base . Then
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.
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