Translation examples
Such a function, and its analytic continuations, is called the hypergeometric function.
Jos sarja suppenee, sen summaa kutsutaan hypergeometriseksi funktioksi.
The Jacobi polynomials are defined via the hypergeometric function as follows: P n ( α , β ) ( z ) = ( α + 1 ) n n ! 2 F 1 ( − n , 1 + α + β + n ; α + 1 ; 1 2 ( 1 − z ) ) , {\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n}}{n!}}\,{}_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\tfrac {1}{2}}(1-z)\right),} where ( α + 1 ) n {\displaystyle (\alpha +1)_{n}} is Pochhammer's symbol (for the rising factorial).
Ne saadaan hypergeometrisista sarjoista, missä sarjasta otetaan mukaan vain äärellisen monta termiä: P n ( α , β ) ( z ) = ( α + 1 ) n n ! 2 F 1 ( − n , 1 + α + β + n ; α + 1 ; 1 − z 2 ) , {\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n}}{n!}}\,_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\frac {1-z}{2}}\right),} missä ( α + 1 ) n {\displaystyle (\alpha +1)_{n}} on Pochhammerin symboli), (Abramowitz & Stegun p561.) ja siten sillä on olemassa eksplisiittinen lauseke P n ( α , β ) ( z ) = Γ ( α + n + 1 ) n !
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