Hypergeometric series

In mathematics, a hypergeometric series is the sum of a sequence of terms in which the ratios of successive coefficients k is a rational function of k. The series, if convergent, will define a hypergeometric function which may then be defined over a wider domain of the argument by analytic continuation. The hypergeometric series is generally written:

Related Topics:
Mathematics - Coefficient - Rational function - Analytic continuation

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:,_pF_q(a_1,ldots,a_p;b_1,ldots,b_q;x)=sum_{k=0}^infty c_k x^k

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where c_0=1 and

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:rac{c_{k+1}}{c_k}=rac{(k+a_1)(k+a_2)cdots(k+a_p)}{(k+b_1)(k+b_2)cdots(k+b_q)},rac{1}{k+1}

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The series may also be written:

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:,_pF_q(a_1,ldots,a_p;b_1,ldots,b_q;x)=sum_{k=0}^infty

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rac{(a_1)_k(a_2)_kldots(a_p)_k}{(b_1)_k(b_2)_kldots(b_q)_k},rac{x^k}{k!}

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where (a)_k=a(a+1)ldots(a+k-1) is the rising factorial or Pochhammer symbol.

Related Topics:
Rising factorial - Pochhammer symbol

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