Alternating Ordinary Generating Functions

SPECIES

Alternate Simplicial Polytopic Numbers

$\displaystyle \dfrac{1}{1+x} = \sum_{n \geq 0} (-1)^n \, x^n = 1 - x + x^2 - x^3 + x^4 - x^5 \pm \cdots$

$\displaystyle \dfrac{1}{(1+x)^2} = \sum_{n \geq 1} (-1)^{n-1} \, n \, x^{n-1} = 1 - 2x + 3x^2 - 4x^3 + 5x^4 \pm \cdots$

$\displaystyle \dfrac{1}{(1+x)^3} = \sum_{n \geq 2} (-1)^n \, \dfrac{n(n-1)}{2} \, x^{n-2} \\ = 1 - 3x + 6x^2 - 10x^3 + 15x^4 \pm \cdots$

$\displaystyle \dfrac{1}{(1+x)^4} = \sum_{n \geq 3} (-1)^{n-1} \, \dfrac{n(n-1)(n-2)}{6} \, x^{n-3} \\ = 1 - 4x + 10x^2 - 20x^3 + 35x^4 \pm \cdots$

$\displaystyle \dfrac{1}{(1+x)^5} = \sum_{n \geq 4} (-1)^n \, \dfrac{n(n-1)(n-2)(n-3)}{24} \, x^{n-4} \\ = 1 - 5x + 15x^2 - 35x^3 + 70x^4 \pm \cdots$

$\displaystyle \dfrac{1}{(1+x)^6} = \sum_{n \geq 5} (-1)^{n-1} \, \dfrac{n(n-1)(n-2)(n-3)(n-4)}{120} \, x^{n-5} \\ = 1 - 6x + 21x^2 - 56x^3 + 126x^4 \pm \cdots$

Alternate Square Roots

$\displaystyle \sqrt{1+x} = \sum_{n \geq 0} (-1)^{n-1} \dfrac{(2n)!}{(2n-1)\,(2^n\,n!)^2} \, x^n \\ \\ = 1 + \dfrac{1}{2} x - \dfrac{1}{8} x^2 + \dfrac{1}{16} x^3 \pm \cdots$

$\displaystyle \sqrt{1+x^2} = \sum_{n \geq 0} (-1)^{n-1} \dfrac{(2n)!}{(2n-1)\,(2^n\,n!)^2} \, x^{2n} \\ \\ = 1 + \dfrac{1}{2} x^2 - \dfrac{1}{8} x^4 + \dfrac{1}{16} x^6 \pm \cdots$

$\displaystyle \dfrac{1}{\sqrt{1+x}} = \sum_{n \geq 0} (-1)^n \dfrac{(2n)!}{(2^n\,n!)^2} \, x^n \\ \\ = 1 - \dfrac{1}{2} x + \dfrac{3}{8} x^2 - \dfrac{5}{16} x^3 \pm \cdots$

$\displaystyle \dfrac{1}{\sqrt{1+x^2}} = \sum_{n \geq 0} (-1)^n \dfrac{(2n)!}{(2^n\,n!)^2} \, x^{2n} \\ \\ = 1 - \dfrac{1}{2} x^2 + \dfrac{3}{8} x^4 - \dfrac{5}{16} x^6 \pm \cdots$

Alternate Circular Trigonometric Functions

$\displaystyle \sin x = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(2n+1)!} \, x^{2n+1}$

$\displaystyle \cos x = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(2n)!} \, x^{2n}$

$\displaystyle \tan x = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}\,(2^{2n}-1)}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \csc x = -\sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}-2}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \sec x = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(2n)!} \, E_{2n} \, x^{2n}$

$\displaystyle \cot x = \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \sin\dfrac{x}{2} = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{2^{2n+1}\,(2n+1)!} \, x^{2n+1}$

$\displaystyle \cos\dfrac{x}{2} = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{2^{2n}\,(2n)!} \, x^{2n}$

$\displaystyle \tan\dfrac{x}{2} = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2(2^{2n}-1)}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \csc\dfrac{x}{2} = -\sum_{n \geq 0} (-1)^n \, \dfrac{2(1-2^{1-2n})}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \sec\dfrac{x}{2} = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{2^{2n}\,(2n)!} \, E_{2n} \, x^{2n}$

$\displaystyle \cot\dfrac{x}{2} = \sum_{n \geq 0} (-1)^n \, \dfrac{2}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \sin 2x = \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n+1}}{(2n+1)!} \, x^{2n+1}$

$\displaystyle \cos 2x = \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}}{(2n)!} \, x^{2n}$

$\displaystyle \tan 2x = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{4n-1}\,(2^{2n}-1)}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \csc 2x = -\sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}\,(2^{2n-1}-1)}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle \sec 2x = \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}}{(2n)!} \, E_{2n} \, x^{2n}$

$\displaystyle \cot 2x = \sum_{n \geq 0} (-1)^n \, \dfrac{2^{4n-1}}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle x\sin x = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(2n+1)!} \, x^{2n+2}$

$\displaystyle x\cos x = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(2n)!} \, x^{2n+1}$

$\displaystyle x\tan x = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}\,(2^{2n}-1)}{(2n)!} \, B_{2n} \, x^{2n}$

$\displaystyle x\csc x = \sum_{n \geq 0} (-1)^{n-1} \, \dfrac{2^{2n}-2}{(2n)!} \, B_{2n} \, x^{2n}$

$\displaystyle x\sec x = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(2n)!} \, E_{2n} \, x^{2n+1}$

$\displaystyle x\cot x = \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}}{(2n)!} \, B_{2n} \, x^{2n}$

$\displaystyle x\sin x + \cos x = \sum_{n \geq 0} (-1)^{n-1} \, \dfrac{2n-1}{(2n)!} \, x^{2n}$

$\displaystyle x\sin x - \cos x = \sum_{n \geq 0} (-1)^{n-1} \, \dfrac{2n+1}{(2n)!} \, x^{2n}$

$\displaystyle x\cos x + \sin x = \sum_{n \geq 0} (-1)^n \, \dfrac{2n+2}{(2n+1)!} \, x^{2n+1}$

$\displaystyle x\cos x - \sin x = \sum_{n \geq 0} (-1)^n \, \dfrac{2n}{(2n+1)!} \, x^{2n+1}$

$\displaystyle \dfrac{1}{\cos x \, \sin x} = \sec x \, \csc x = 2\csc(2x) = -\sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n+1}(2^{2n-1}-1)}{(2n)!} \, B_{2n} \, x^{2n-1}$

$\displaystyle 1 - \dfrac{x}{2} \cot\dfrac{x}{2} = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{1}{(2n)!} \, B_{2n} \, x^{2n}$

$\displaystyle \sin^2 x = \dfrac{1}{2} - \dfrac{1}{2} \, \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}}{(2n)!} \, x^{2n}$

$\displaystyle \cos^2 x = \dfrac{1}{2} + \dfrac{1}{2} \, \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}}{(2n)!} \, x^{2n}$

$\displaystyle \tan^2 x = -1 \,+\, \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}\,(2^{2n}-1)}{(2n-2)!} \, \dfrac{B_{2n}}{2n} \, x^{2n-2}$

$\displaystyle \csc^2 x = \cot^2 x +1 = \sum_{n \geq 0} (-1)^{n-1} \, \dfrac{2^{2n}\,(2n-1)}{(2n)!} \, B_{2n} \, x^{2n-2}$

$\displaystyle \sec^2 x = \tan^2 x +1 = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}(2^{2n}-1)}{(2n-2)!} \, \dfrac{B_{2n}}{2n} \, x^{2n-2}$

$\displaystyle \cot^2 x = \csc^2 x - 1 = -1 \,-\, \sum_{n \geq 0} (-1)^n \, \dfrac{2^{2n}(2n-1)}{(2n)!} \, B_{2n} \, x^{2n-2}$

Alternate Inverse Circular Trigonometric Functions

$\displaystyle \arctan x = \sum_{n \geq 0} (-1)^n \, \dfrac{x^{2n+1}}{2n+1}$

Alternate Inverse Hyperbolic Trigonometric Functions

$\displaystyle \text{arsinh}\,x = \text{arcsch}\,\dfrac{1}{x} = \ln(x+\sqrt{1+x^2}) \\ \\ = \sum_{n \geq 0} (-1)^n \, \dfrac{(2n)!}{2^{2n}(2n+1)(n!)^2} \, x^{2n+1} = x - \dfrac{x^3}{6} + \dfrac{3x^5}{40} - \dfrac{5x^7}{112} \pm \cdots$

$\displaystyle \text{arsinh}\,x - \ln(2x) = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{(2n\!-\!1)!!}{2n(2n)!!} \, \dfrac{1}{x^{2n}} \\ \\ = -\ln x + \dfrac{1}{4x^2} - \dfrac{3}{32x^4} + \dfrac{5}{96x^6} - \dfrac{35}{1024x^8} \pm \cdots$

$\displaystyle \text{arcsch}\,x = \text{arsinh}\,\dfrac{1}{x} = \ln\dfrac{1+\sqrt{1+x^2}}{x} = \ln(1+\sqrt{1+x^2}) - \ln x \\ \\ = \sum_{n \geq 0} (-1)^n \dfrac{(2n)!}{2^{2n}(2n+1)(n!)^2} \, \dfrac{1}{x^{2n+1}} = \dfrac{1}{x} - \dfrac{1}{6x^3} + \dfrac{3}{40x^5} - \dfrac{5}{112x^7} \pm \nolinebreak \cdots$

Alternate Fresnel Integrals

$\displaystyle S(x) = \int_0^x \sin t^2 \, dt = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(4n+3)(2n+1)!} \, x^{4n+3}$

$\displaystyle C(x) = \int_0^x \cos t^2 \, dt = \sum_{n \geq 0} (-1)^n \, \dfrac{1}{(4n+1)(2n)!} \, x^{4n+1}$

Alternate Exponential Functions

$\displaystyle e^{-x} = \sum_{n \geq 0} (-1)^{n} \, \dfrac{x^n}{n!}$

$\displaystyle e^{e^{-x}-1} = \sum_{n \geq 0} (-1)^{n} \, \dfrac{B_n}{n!} \, x^n \qquad (B_n \text{ is the n-th Bell number)}$

Alternate Logarithmic Functions

$\displaystyle \ln x = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{(x-1)^n}{n}$

$\displaystyle \ln(1+x) = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{x^n}{n}$

$\displaystyle \ln(1+x^2) = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{x^{2n}}{n}$

$\displaystyle \ln\sqrt{1+x^2} = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{x^{2n}}{2n}$

Alternate Logarithmic Trigonometric Functions

$\displaystyle \ln(\sin x) = \ln x - \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(\cos x) = -\sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}(2^{2n}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(\tan x) = \ln x + \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n+1}(2^{2n-1}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(\csc x) = -\ln(\sin x) = -\ln x + \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(\sec x) = -\ln(\cos x)= \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}(2^{2n}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(\cot x) = -\ln(\tan x) = -\ln x \,-\, \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n+1}(2^{2n-1}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\sin\dfrac{x}{2}\right) = \ln x - \ln 2 - \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{1}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\cos\dfrac{x}{2}\right) = -\sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}-1}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\tan\dfrac{x}{2}\right) = \ln x - \ln 2 + \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2(2^{2n-1}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\csc\dfrac{x}{2}\right) = -\ln\!\left(\sin\dfrac{x}{2}\right) = \ln 2 - \ln x + \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{1}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\sec\dfrac{x}{2}\right) = -\ln\!\left(\cos\dfrac{x}{2}\right)= \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}-1}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\cot\dfrac{x}{2}\right) = -\ln\!\left(\tan\dfrac{x}{2}\right) = \ln 2 - \ln x \,-\, \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2(2^{2n-1}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\dfrac{\sin x}{x}\right) = -\sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\dfrac{\cos x}{x}\right) = -\ln x - \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}(2^{2n}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln\!\left(\dfrac{\tan x}{x}\right) = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n+1}(2^{2n-1}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(x\csc x) = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(x\sec x) = -\ln x + \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}(2^{2n}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(x\cot x) = -\sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n+1}(2^{2n-1}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

$\displaystyle \ln(x(1+\sec x)) = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{2^{2n}(2^{2n}-1)}{(2n)!} \, \dfrac{B_{2n}}{2n} \, x^{2n}$

Alternate Repeated Numbers

$\displaystyle \dfrac{1}{(1+10x)(1+x)} = \sum_{n \geq 0} (-1)^n \left(\sum_{m=0}^n 10^m\right) x^n \\ \\ = 1 - 11x + 111x^3 - 1111x^4 \pm \cdots$

$\displaystyle \dfrac{1}{(1+x)(1-x^2)} = \dfrac{1}{(1-x)(1+x)^2} = \sum_{n \geq 1} (-1)^{n-1} \left\lfloor\dfrac{n+1}{2}\right\rfloor x^{n-1} \\ \\ = 1 - x + 2x^2 - 2x^3 + 3x^4 - 3x^5 \pm \cdots$

$\displaystyle \dfrac{1}{(1+10x)(1+x)^2} = \sum_{n \geq 0} (-1)^n \left(\dfrac{10(10^{n+1}-1)}{81} - \dfrac{n+1}{9}\right) x^n \\ \\ = 1 - 12x + 123x^3 - 1234x^4 \pm \cdots$

Alternate Zeta Function

$\displaystyle \dfrac{1}{2} + x\,\psi^{(0)}(x) = \sum_{n \geq 0} (-1)^n \, \zeta(n) \, x^n$

$\displaystyle -\psi^{(0)}(x+1) = -\dfrac{1}{x} - \psi^{(0)}(x) = \gamma - \dfrac{1}{x} - H_{x-1} = \sum_{n \geq 1} (-1)^{n-1} \, \zeta(n) \, x^{n-1}$

$\displaystyle \dfrac{1}{x} \left(\gamma + \psi^{(0)}(x+1)\right) = \dfrac{1}{x^2} + \dfrac{\gamma}{x} + \dfrac{\psi^{(0)}(x)}{x} = \dfrac{1}{x^2} + \dfrac{H_{x-1}}{x} = \sum_{n \geq 2} (-1)^n \, \zeta(n) \, x^{n-2}$

$\displaystyle \ln\dfrac{1}{\Gamma(x+1)} = -\ln\Gamma(x+1) = \sum_{n \geq 1} (-1)^{n-1} \, \dfrac{\zeta(n)}{n} \, x^n$

$\displaystyle \gamma x + \ln\Gamma(x+1) = \sum_{n \geq 2} (-1)^n \, \dfrac{\zeta(n)}{n} \, x^n$

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