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## Homework Statement

[tex]_2F_1(a,b;c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{(c)_nn!}x^n [/tex]

Show that Legendre polynomial of degree ##n## is defined by

[tex]P_n(x)=\,_2F_1(-n,n+1;1;\frac{1-x}{2})[/tex]

## Homework Equations

Definition of Pochamer symbol[/B]

[tex](a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}[/tex]

## The Attempt at a Solution

[tex]P_n(x)=\,_2F_1(-n,n+1;1;\frac{1-x}{2})=\sum^{\infty}_{k=0}\frac{\frac{\Gamma(-n+k)}{\Gamma(-n)}\frac{\Gamma(n+k+1)}{\Gamma(n+1)}}{(k!)^2}\frac{(1-x)^k}{2^k}[/tex]

Main problem for me is ##\Gamma(-n)##. Bearing in mind that ##n## is degree of polynomial, ##\Gamma(-n)## diverge. What is solution of this issue?