My Math Forum a question about a power series

 Complex Analysis Complex Analysis Math Forum

 May 15th, 2011, 10:58 AM #1 Senior Member   Joined: Aug 2008 Posts: 133 Thanks: 0 a question about a power series If we let $x \bigtriangleup y= \ln(e^x+e^y)$ be deltation, its commutative and associative, we find addition is distributive across it: $(x \bigtriangleup y)+l= (x+l) \bigtriangleup (y+l)$, and that it is defined by the following recursion formula: $x_1 \bigtriangleup x_2 \bigtriangleup ... x_n= x + \ln(n)$, multiplication across deltation also behaves as exponentiation across addition. So the question I had was, what's its taylor series? I managed to work out the first five derivatives, I'm gonna leave out the steps, I just wonder if someone can maybe help me out with the algorithm or if they've seen these coefficients somewhere before (or maybe someone knows a technique for iterated differentiation): $x \bigtriangleup y = ln(e^x + e^y)\\ \frac{d}{dx} x \bigtriangleup y = \frac{1}{1+e^{y-x}}\\ \frac{d^2}{dx^2} x \bigtriangleup y = \frac{e^{y-x}}{(1+e^{y-x})^2}\\ \frac{d^3}{dx^3} x \bigtriangleup y = 2\frac{e^{2y-2x}}{(1+e^{y-x})^3} - \frac{e^{y-x}}{(1+e^{y-x})^2}\\ \frac{d^4}{dx^4} x \bigtriangleup y = 6\frac{e^{3y-3x}}{(1+e^{y-x})^4} - 6\frac{e^{2y-2x}}{(1+e^{y-x})^3} + \frac{e^{y-x}}{(1+e^{y-x})^2}\\ \frac{d^5}{dx^5} x \bigtriangleup y = 24\frac{e^{4y-4x}}{(1+e^{y-x})^5} - 36\frac{e^{3y-3x}}{(1+e^{y-x})^4} + 14\frac{e^{2y-2x}}{(1+e^{y-x})^3} - \frac{e^{y-x}}{(1+e^{y-x})^2}$ Interestingly enough, the coefficients always add up to one. The obvious choice is to center the taylor series about y, so if $f(x)= x \bigtriangleup y$ $f(y) = y \bigtriangleup y = y + ln(2)\\ f'(y) = \frac{1}{2}\\ f''(y) = \frac{1}{4}\\ f^{(3)}(y) = 0\\ f^{(4)}(y) = -\frac{1}{8}\\ f^{(5)}(y) = 0$ therefore the taylor series will look something like this: $x \bigtriangleup y= y + \ln(2) + \sum_{n=1}^{\infty} a_n (x-y)^n$ where $a_1= \frac{1}{2},\, a_2 = \frac{1}{8},\, a_3 = 0...$ etc... So thanks for reading if you got this far, any help would be greatly appreciated.

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### deltation math

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