{"history":[],"article":{"id":"8efea134-a806-4802-b8d2-38cca48deacc","title":"Appell series","slug":"appell-series","content":":::summary\n:::\n\n# \nIn mathematics, **Appell [series](https://bliptext.com/articles/series-mathematics)** are a set of four hypergeometric [series](https://bliptext.com/articles/series-mathematics) F1, F2, F3, F4 of two [variables](https://bliptext.com/articles/variable-mathematics) that were introduced by and that generalize Gauss's hypergeometric [series](https://bliptext.com/articles/series-mathematics) 2F1 of one variable.\nAppell established the set of [partial differential equations](https://bliptext.com/articles/partial-differential-equation) of which these [functions](https://bliptext.com/articles/function-mathematics) are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.\n\n# Definitions\nThe Appell series F1 is defined for |x| < 1, |y| < 1 by the double series\nwhere (q)_n is the [Pochhammer symbol](https://bliptext.com/articles/pochhammer-symbol).\nFor other values of x and y the function F1 can be defined by [analytic continuation](https://bliptext.com/articles/analytic-continuation).\nIt can be shown that\nSimilarly, the function F2 is defined for |x| + |y| < 1 by the series\nand it can be shown that\nAlso the function F3 for |x| < 1, |y| < 1 can be defined by the series\nand the function F4 for |x|1/2 + |y|1/2 < 1 by the series\n\n# Recurrence relations\nLike the Gauss hypergeometric series 2F1, the Appell double series entail [recurrence relations](https://bliptext.com/articles/recurrence-relation) among contiguous functions.\nFor example, a basic set of such relations for Appell's F1 is given by:\nAny other relation valid for F1 can be derived from these four.\nSimilarly, all recurrence relations for Appell's F3 follow from this set of five:\n\n# Derivatives and differential equations\nFor Appell's F1, the following [derivatives](https://bliptext.com/articles/derivative) result from the definition by a double series:\nFrom its definition, Appell's F1 is further found to satisfy the following system of second-order [differential equations](https://bliptext.com/articles/partial-differential-equation):\nA system partial differential equations for F2 is\nThe system have solution\nSimilarly, for F3 the following derivatives result from the definition:\nAnd for F3 the following system of differential equations is obtained:\nA system partial differential equations for F4 is\nThe system has solution\n\n# Integral representations\nThe four functions defined by Appell's double series can be represented in terms of [double integrals](https://bliptext.com/articles/multiple-integral) involving [elementary functions](https://bliptext.com/articles/elementary-functions) only.\nHowever, discovered that Appell's F1 can also be written as a one-dimensional [Euler](https://bliptext.com/articles/leonhard-euler)-type [integral](https://bliptext.com/articles/integral):\nThis representation can be verified by means of [Taylor expansion](https://bliptext.com/articles/taylor-series) of the integrand, followed by termwise integration.\n\n# Special cases\nPicard's integral representation implies that the [incomplete elliptic integrals](https://bliptext.com/articles/elliptic-integral) F and E as well as the [complete elliptic integral](https://bliptext.com/articles/elliptic-integral) Π are special cases of Appell's F1:\n\n# Related series","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:12:21.311Z","updated_at":"2025-01-26T11:12:21.311Z","search_vector":"'appel':1 'seri':2","search_content":" \n \n\n# \nIn mathematics, **Appell [series](https://bliptext.com/articles/series-mathematics)** are a set of four hypergeometric [series](https://bliptext.com/articles/series-mathematics) F1, F2, F3, F4 of two [variables](https://bliptext.com/articles/variable-mathematics) that were introduced by and that generalize Gauss's hypergeometric [series](https://bliptext.com/articles/series-mathematics) 2F1 of one variable.\nAppell established the set of [partial differential equations](https://bliptext.com/articles/partial-differential-equation) of which these [functions](https://bliptext.com/articles/function-mathematics) are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.\n\n# Definitions\nThe Appell series F1 is defined for |x| < 1, |y| < 1 by the double series\nwhere (q)_n is the [Pochhammer symbol](https://bliptext.com/articles/pochhammer-symbol).\nFor other values of x and y the function F1 can be defined by [analytic continuation](https://bliptext.com/articles/analytic-continuation).\nIt can be shown that\nSimilarly, the function F2 is defined for |x| + |y| < 1 by the series\nand it can be shown that\nAlso the function F3 for |x| < 1, |y| < 1 can be defined by the series\nand the function F4 for |x|1/2 + |y|1/2 < 1 by the series\n\n# Recurrence relations\nLike the Gauss hypergeometric series 2F1, the Appell double series entail [recurrence relations](https://bliptext.com/articles/recurrence-relation) among contiguous functions.\nFor example, a basic set of such relations for Appell's F1 is given by:\nAny other relation valid for F1 can be derived from these four.\nSimilarly, all recurrence relations for Appell's F3 follow from this set of five:\n\n# Derivatives and differential equations\nFor Appell's F1, the following [derivatives](https://bliptext.com/articles/derivative) result from the definition by a double series:\nFrom its definition, Appell's F1 is further found to satisfy the following system of second-order [differential equations](https://bliptext.com/articles/partial-differential-equation):\nA system partial differential equations for F2 is\nThe system have solution\nSimilarly, for F3 the following derivatives result from the definition:\nAnd for F3 the following system of differential equations is obtained:\nA system partial differential equations for F4 is\nThe system has solution\n\n# Integral representations\nThe four functions defined by Appell's double series can be represented in terms of [double integrals](https://bliptext.com/articles/multiple-integral) involving [elementary functions](https://bliptext.com/articles/elementary-functions) only.\nHowever, discovered that Appell's F1 can also be written as a one-dimensional [Euler](https://bliptext.com/articles/leonhard-euler)-type [integral](https://bliptext.com/articles/integral):\nThis representation can be verified by means of [Taylor expansion](https://bliptext.com/articles/taylor-series) of the integrand, followed by termwise integration.\n\n# Special cases\nPicard's integral representation implies that the [in[complete elliptic integral](https://bliptext.com/articles/elliptic-integral)s](https://bliptext.com/articles/elliptic-integral) F and E as well as the [complete elliptic integral](https://bliptext.com/articles/elliptic-integral) Π are special cases of Appell's F1:\n\n# Related series","revisionCount":0},"hasMore":false,"nextCursor":null}

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