P-adic gamma function
In mathematics, the "p"-adic gamma function Γ is a function of a "p"-adic variable analogous to the gamma function. It was first explicitly defined by , though pointed out that implicitly used the same function. defined a "p"-adic analog "G" of log Γ. 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. Definition. The "p"-adic gamma function is the unique continuous function of a "p"-adic integer "x" (with values in formula_1) 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 formula_1, formula_3 can be extended uniquely to the whole of formula_1. Here formula_1 is the ring of "p"-adic integers. It follows from the definition that the values of formula_6 are invertible in formula_1; this is because these values are products of integers not divisible by "p", and this property holds after the continuous extension to formula_1. Thus formula_9. Here formula_10 is the set of invertible "p"-adic integers. Basic properties of the p-adic gamma function. The classical  gamma function satisfies the functional equation formula_11 for any formula_12. This has an analogue with respect to the Morita gamma function: The Euler's reflection formula formula_14 has its following simple counterpart in the "p"-adic case: where formula_16 is the first digit in the "p"-adic expansion of "x", unless formula_17, in which case formula_18 rather than 0. Special values. and, in general, At formula_24 the Morita gamma function is related to the Legendre symbol formula_25: It can also be seen, that formula_27 hence formula_28 as formula_29. 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. For example, where formula_32 denotes the square root with first digit 3, and formula_33 denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.) Another example is where formula_35 is the square root of formula_36 in formula_37 congruent to 1 modulo 3. Section::::"p"-adic Raabe formula. The Raabe-formula for the classical Gamma function says that This has an analogue for the Iwasawa logarithm of the Morita gamma function: The ceiling function to be understood as the "p"-adic limit formula_40 such  that formula_41 through rational integers. Mahler expansion. 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: where the sequence formula_43 is defined by the following identity: