THE GAMMA FUNCTION (FACTORIAL FUNCTION)

The aim of this note is to obtain a generalization of a very simple, elegant but powerful convergence lemma introduced by Mortici [Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comp., 215 (2010), No. 11, 4044–4048; Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), No. 5, 434–441; Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93 (2009), No. 1, 37–45; Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), No. 2, 186–191] and exploited by him and other authors in an impressive number of recent and very recent papers devoted to constructing asymptotic expansions, accelerating famous sequences in mathematics, developing approximation formulas for factorials that improve various classical results etc. We illustrate the new result by some important particular cases and also indicate a way for using it in similar contexts.

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A Study on Novel Extensions for the $p$-adic Gamma and $p$-adic Beta Functions

10.20944/preprints201905.0261.v1 ◽

2019 ◽

Author(s):

Ugur Duran ◽

Mehmet Acikgoz

Keyword(s):

Gamma Function ◽

Beta Function ◽

Euler Constant ◽

Factorial Function

In this paper, we introduce the (ρ,q)-analogue of the p-adic factorial function. By utilizing some properties of (ρ,q)-numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic (ρ,q)-gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2. We also derive more representations of the p-adic (ρ,q)-gamma function in general case. Moreover, we consider the p-adic (ρ,q)-Euler constant derived from the derivation of p-adic (ρ,q)-gamma function at x = 1. Furthermore, we provide a limit representation of aforementioned Euler constant based on (ρ,q)-numbers. Finally, we consider (ρ,q)-extension of the p-adic beta function via the p-adic (ρ,q)-gamma function and we then investigate various formulas and identities.

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A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions

Mathematical and Computational Applications ◽

10.3390/mca24020053 ◽

2019 ◽

Vol 24 (2) ◽

pp. 53

Author(s):

Ugur Duran ◽

Mehmet Acikgoz

Keyword(s):

Gamma Function ◽

Beta Function ◽

Euler Constant ◽

Factorial Function

In this paper, we introduce the ρ , q -analog of the p-adic factorial function. By utilizing some properties of ρ , q -numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic ρ , q -gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2 . We also derive more representations of the p-adic ρ , q -gamma function in general case. Moreover, we consider the p-adic ρ , q -Euler constant derived from the derivation of p-adic ρ , q -gamma function at x = 1 . Furthermore, we provide a limit representation of aforementioned Euler constant based on ρ , q -numbers. Finally, we consider ρ , q -extension of the p-adic beta function via the p-adic ρ , q -gamma function and we then investigate various formulas and identities.

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A fresh look at Euler's limit formula for the gamma function

The Mathematical Gazette ◽

10.1017/s0025557200001261 ◽

2014 ◽

Vol 98 (542) ◽

pp. 235-242 ◽

Cited By ~ 3

Author(s):

G. J. O. Jameson

Keyword(s):

Continuous Function ◽

Gamma Function ◽

Smoothness Condition ◽

Limit Formula ◽

Factorial Function ◽

Definition Of ◽

The Right

Consider the problem of defining a continuous function f(x) which agrees with factorials at integers. There are many possible ways to do this. In fact, such a function can be constructed by taking any continuous definition of f(x) on [0,1] with f(0) = f(1) = 1 (such as f(x) = 1), and then extending the definition to all x > 1 by the formula f(x + 1) = (x + 1)f(x). This construction was discussed by David Fowler in [1] and [2]. For example, the choice f(x) = ½x(x − 1) + 1 results in a function that is differentiable everywhere, including at integers.However, this approach had already been overtaken in 1729, when Euler obtained the conclusive solution to the problem by defining what we now call the gamma function. Among all the possible functions that reproduce factorials, this is the ‘right’ one, in the sense that it is the only one satisfying a certain smoothness condition which we will specify below. Admittedly, Euler didn't know this. It is known as the Bohr-Mollerup theorem, and was only proved nearly two centuries later.First, a remark on notation: the notation Γ (x) for the gamma function, introduced by Legendre, is such that Γ (n) is actually (n − 1)! instead of n!. Though this might seem a little perverse, it does result in some formulae becoming slightly neater. Some writers, including Fowler, write x! for Γ (x + 1), and refer to this as the ‘factorial function’. However, the notation Γ (x) is very firmly entrenched, and I will adhere to it here.

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