mersenneforum.org > Math Derivative of Gamma
 User Name Remember Me? Password
 Register FAQ Search Today's Posts Mark Forums Read

 2008-01-08, 05:53 #1 nibble4bits     Nov 2005 181 Posts Derivative of Gamma If the convention is used that x!=Gamma(x+1), then the factorial function can have all real numbers except negative integers as inputs. Thinking about this lead me to a path close to the reasoning behind Stirling's approximation of factorial. He used ln 0 + ln 1 + ln 2 + ln 3 + ln 4 + ... to derive his approximation. I was looking at functions like x^x, and trying to create a sequence in place of the power: x^g_n(x) where g_0(x) = x and each next g(x) gives a closer approximation. The derivative is x!*Digamma(x+1). In other words: Dx x! = Gamma(x+1) * Digamma(x+1) x! * Digamma(x+1) = x!* Dx Ln(x!) Dx Digamma(x+1) = Zeta(2, x+1) 3rd derivative of x! = x!*(Digamma(x+1)^3 + 3*Zeta(2, x+1)*Digamma(x+1) - 2*Zeta(3, x+1)) According to Derive, this is the derivative of that previous line without the x! in the product: 3·Zeta(2, x + 1)^2·Digamma(x + 1) - 6·Zeta(3, x + 1)·Digamma(x + 1) + 6·Zeta(4, x + 1) + 3·Zeta(2, x + 1) There seems to be a chain: Gamma -> Digamma -> Zeta I'm wondering if there's a formula to get the n'th derivative of x!, past the first one containing the Zeta function. Note: I'm using the convention that allows factorial to take all real numbers except negative integers as inputs. ('dependant variables' if I remember right) This problem is connected to the third one I'm thinking of. I need a derivative of (-1)^x and x! before I can continue.

 Thread Tools

 Similar Threads Thread Thread Starter Forum Replies Last Post Calvin Culus Analysis & Analytic Number Theory 6 2010-12-23 22:18 jinydu Math 2 2008-11-24 10:22

All times are UTC. The time now is 06:14.

Wed Nov 25 06:14:13 UTC 2020 up 76 days, 3:25, 4 users, load averages: 0.87, 1.30, 1.44

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.