# special cases of hypergeometric function

Many elementary (http://planetmath.org/ElementaryFunction) and non-elementary transcendental functions may be expressed as special cases of the hypergeometric functions

 $F(a,\,b,\,c;\,x)\;=\;1+\frac{ab}{1!c}x+\frac{a(a+1)b(b+1)}{2!c(c+1)}x^{2}+% \frac{a(a+1)(a+2)b(b+1)(b+2)}{3!c(c+1)(c+2)}x^{3}+\ldots,$

which are solutions of the hypergeometric equation

 $x(x-1)\frac{d^{2}y}{dx^{2}}+(c-(a+b+1))\frac{dy}{dx}-aby\;=\;0.$

For example:

• $(1\!+\!x)^{n}\;=\;F(-n,\,1,\,1;\,-x)$

• $\ln(1\!+\!x)\;=\;xF(1,\,1,\,2;\,-x)$

• $\ln\frac{1+x}{1-x}\;=\;2xF(\frac{1}{2},\,1,\,\frac{3}{2};\,x^{2})$

• $\arcsin{x}\;=\;xF(\frac{1}{2},\,\frac{1}{2},\,\frac{3}{2};\,x^{2})$

• $\arctan{x}\;=\;xF(\frac{1}{2},\,1,\,\frac{3}{2};\,-x^{2})$

• $\sin(m\arcsin{x})\;=\;mxF(\frac{1+m}{2},\,\frac{1-m}{2},\,\frac{3}{2};\,x^{2})$

• $\cos(m\arcsin{x})\;=\;F(\frac{m}{2},\,-\frac{m}{2},\,\frac{1}{2};\,x^{2})$

• $T_{n}(x)\;=\;F(n,\,-n,\,\frac{1}{2};\,\frac{1-x}{2})$  (Chebyshev polynomials)

• $P_{n}(x)\;=\;F(-n,\,n+1,\,1;\,\frac{1-x}{2})$  (Legendre polynomials)

• $\displaystyle\int_{0}^{\frac{\pi}{2}}\!\frac{d\varphi}{\sqrt{1\!-\!x^{2}\sin^{% 2}\varphi}}$ $\;=\;\frac{\pi}{2}F(\frac{1}{2},\,\frac{1}{2},\,1;\,x^{2})$  (complete elliptic integral of 1st kind)

• $\displaystyle\int_{0}^{\frac{\pi}{2}}\!\sqrt{1\!-\!x^{2}\sin^{2}\varphi}\;d\varphi$ $\;=\;\frac{\pi}{2}F(-\frac{1}{2},\,\frac{1}{2},\,1;\,x^{2})$  (complete elliptic integral of 2nd kind)

Title special cases of hypergeometric function SpecialCasesOfHypergeometricFunction 2013-03-22 18:54:39 2013-03-22 18:54:39 pahio (2872) pahio (2872) 8 pahio (2872) Example msc 33C05 FrobeniusMethod IndexOfSpecialFunctions GettingTaylorSeriesFromDifferentialEquation