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Title: Generalized Hypergeometric Function -- from Wolfram MathWorld

Open Graph Title: Generalized Hypergeometric Function -- from Wolfram MathWorld

X Title: Generalized Hypergeometric Function -- from Wolfram MathWorld

Description: The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written (c_(k+1))/(c_k)=(P(k))/(Q(k))=((k+a_1)(k+a_2)...(k+a_p))/((k+b_1)(k+b_2)...(k+b_q)(k+1)). (1) (The factor of k+1 in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written sum_(k=0)^(infty)c_kx^k = _pF_q[a_1,a_2,...,a_p; b_1,b_2,...,b_q;x] (2) =...

Open Graph Description: The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written (c_(k+1))/(c_k)=(P(k))/(Q(k))=((k+a_1)(k+a_2)...(k+a_p))/((k+b_1)(k+b_2)...(k+b_q)(k+1)). (1) (The factor of k+1 in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written sum_(k=0)^(infty)c_kx^k = _pF_q[a_1,a_2,...,a_p; b_1,b_2,...,b_q;x] (2) =...

X Description: The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written (c_(k+1))/(c_k)=(P(k))/(Q(k))=((k+a_1)(k+a_2)...(k+a_p))/((k+b_1)(k+b_2)...(k+b_q)(k+1)). (1) (The factor of k+1 in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written sum_(k=0)^(infty)c_kx^k = _pF_q[a_1,a_2,...,a_p; b_1,b_2,...,b_q;x] (2) =...

Opengraph URL: https://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html

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DC.TitleGeneralized Hypergeometric Function
DC.CreatorWeisstein, Eric W.
DC.DescriptionThe generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written (c_(k+1))/(c_k)=(P(k))/(Q(k))=((k+a_1)(k+a_2)...(k+a_p))/((k+b_1)(k+b_2)...(k+b_q)(k+1)). (1) (The factor of k+1 in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written sum_(k=0)^(infty)c_kx^k = _pF_q[a_1,a_2,...,a_p; b_1,b_2,...,b_q;x] (2) =...
DC.Date.Modified2003-05-25
DC.Subject33C20
DC.RightsCopyright 1999-2026 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement.
DC.Formattext/html
DC.Identifierhttps://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html
DC.Languageen
DC.PublisherWolfram Research, Inc.
DC.Relation.IsPartOfhttps://mathworld.wolfram.com/
DC.TypeText
Last-Modified2003-05-25
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asymptotes (2x^3 + 4x^2 - 9)/(3 - x^2)https://www.wolframalpha.com/input/?i=asymptotes+%282x%5E3+%2B+4x%5E2+-+9%29%2F%283+-+x%5E2%29
domino tiling 4 stepshttps://www.wolframalpha.com/input/?i=domino+tiling+4+steps
Generalised Hypergeometric Series.http://www.amazon.com/exec/obidos/ASIN/B0006VV1ME/ref=nosim/ericstreasuretro
Generalized Hypergeometric Functions.http://www.amazon.com/exec/obidos/ASIN/0198535678/ref=nosim/ericstreasuretro
Multiple Hypergeometric Functions and Applications.http://www.amazon.com/exec/obidos/ASIN/0470151900/ref=nosim/ericstreasuretro
Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs.http://www.amazon.com/exec/obidos/ASIN/0470263423/ref=nosim/ericstreasuretro
Concrete Mathematics: A Foundation for Computer Science, 2nd ed.http://www.amazon.com/exec/obidos/ASIN/0201558025/ref=nosim/ericstreasuretro
Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.http://www.amazon.com/exec/obidos/ASIN/0821820230/ref=nosim/ericstreasuretro
https://blog.wolfram.com/2024/01/25/hypergeometric-functions-from-euler-to-appell-and-beyond/https://blog.wolfram.com/2024/01/25/hypergeometric-functions-from-euler-to-appell-and-beyond/
Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities.http://www.amazon.com/exec/obidos/ASIN/3528069503/ref=nosim/ericstreasuretro
A=B.http://www.amazon.com/exec/obidos/ASIN/1568810636/ref=nosim/ericstreasuretro
https://www2.math.upenn.edu/~wilf/AeqB.htmlhttps://www2.math.upenn.edu/~wilf/AeqB.html
Special Functions.http://www.amazon.com/exec/obidos/ASIN/0828402582/ref=nosim/ericstreasuretro
Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences.http://www.amazon.com/exec/obidos/ASIN/0387064826/ref=nosim/ericstreasuretro
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