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| {{dablink|For generalizations of Lambert series see [[Appell–Lerch series]].}}
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| In mathematics, '''Appell [[series (mathematics)|series]]''' are a set of four [[hypergeometric series]] ''F''<sub>1</sub>, ''F''<sub>2</sub>, ''F''<sub>3</sub>, ''F''<sub>4</sub> of two [[variable (mathematics)|variable]]s that were introduced by {{harvs | txt | authorlink= Paul Émile Appell | first= Paul | last= Appell | year= 1880}} and that generalize [[hypergeometric function|Gauss's hypergeometric series]] <sub>2</sub>''F''<sub>1</sub> of one variable. Appell established the set of [[partial differential equation]]s of which these [[function (mathematics)|function]]s are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.
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| ==Definitions==
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| The Appell series ''F''<sub>1</sub> is defined for |''x''| < 1, |''y''| < 1 by the double series:
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| :<math>
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| F_1(a,b_1,b_2,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
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| </math>
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| where the [[Pochhammer symbol]] (''q'')<sub>''n''</sub> represents the rising factorial:
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| :<math>(q)_n = \frac{\Gamma(q+n)}{\Gamma(q)} = q\,(q+1) \cdots (q+n-1) ~.</math>
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| For other values of ''x'' and ''y'' the function ''F''<sub>1</sub> can be defined by [[analytic continuation]].
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| Similarly, the function ''F''<sub>2</sub> is defined for |''x''| + |''y''| < 1 by the series:
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| :<math>
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| F_2(a,b_1,b_2,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b_1)_m (b_2)_n} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~,
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| </math>
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|
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| the function ''F''<sub>3</sub> for |''x''| < 1, |''y''| < 1 by the series:
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| :<math> | |
| F_3(a_1,a_2,b_1,b_2,c;x,y) = \sum_{m,n=0}^\infty \frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
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| </math>
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| and the function ''F''<sub>4</sub> for |''x''|<sup>½</sup> + |''y''|<sup>½</sup> < 1 by the series:
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| :<math>
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| F_4(a,b,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_{m+n}} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~.
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| </math>
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| ==Recurrence relations== | |
| Like the Gauss hypergeometric series <sub>2</sub>''F''<sub>1</sub>, the Appell double series entail [[recurrence relation]]s among contiguous functions. For example, a basic set of such relations for Appell's ''F''<sub>1</sub> is given by:
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| :<math>
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| (a-b_1-b_2) F_1(a,b_1,b_2,c; x,y) - a \,F_1(a+1,b_1,b_2,c; x,y) + b_1 F_1(a,b_1+1,b_2,c; x,y) + b_2 F_1(a,b_1,b_2+1,c; x,y) = 0 ~,
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| </math> | |
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| :<math>
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| c \,F_1(a,b_1,b_2,c; x,y) - (c-a) F_1(a,b_1,b_2,c+1; x,y) - a \,F_1(a+1,b_1,b_2,c+1; x,y) = 0 ~,
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| </math>
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| :<math>
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| c \,F_1(a,b_1,b_2,c; x,y) + c(x-1) F_1(a,b_1+1,b_2,c; x,y) - (c-a)x \,F_1(a,b_1+1,b_2,c+1; x,y) = 0 ~,
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| </math> | |
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| :<math>
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| c \,F_1(a,b_1,b_2,c; x,y) + c(y-1) F_1(a,b_1,b_2+1,c; x,y) - (c-a)y \,F_1(a,b_1,b_2+1,c+1; x,y) = 0 ~.
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| </math>
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| Any other relation valid for ''F''<sub>1</sub> can be derived from these four.
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| Similarly, all recurrence relations for Appell's ''F''<sub>3</sub> follow from this set of five:
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| :<math>
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| c \,F_3(a_1,a_2,b_1,b_2,c; x,y) + (a_1+a_2-c) F_3(a_1,a_2,b_1,b_2,c+1; x,y) - a_1 F_3(a_1+1,a_2,b_1,b_2,c+1; x,y) - a_2 F_3(a_1,a_2+1,b_1,b_2,c+1; x,y) = 0 ~,
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| </math>
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| | |
| :<math>
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| c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1+1,a_2,b_1,b_2,c; x,y) + b_1 x \,F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) = 0 ~,
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| </math>
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| :<math>
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| c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2+1,b_1,b_2,c; x,y) + b_2 y \,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~,
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| </math>
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| :<math>
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| c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2,b_1+1,b_2,c; x,y) + a_1 x \,F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) = 0 ~,
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| </math>
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| :<math>
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| c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2,b_1,b_2+1,c; x,y) + a_2 y \,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~.
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| </math>
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| ==Derivatives and differential equations==
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| For Appell's ''F''<sub>1</sub>, the following [[derivative]]s result from the definition by a double series:
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| :<math>
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| \frac {\partial} {\partial x} F_1(a,b_1,b_2,c; x,y) = \frac {a b_1} {c} F_1(a+1,b_1+1,b_2,c+1; x,y) ~,
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| </math>
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| :<math>
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| \frac {\partial} {\partial y} F_1(a,b_1,b_2,c; x,y) = \frac {a b_2} {c} F_1(a+1,b_1,b_2+1,c+1; x,y) ~.
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| </math>
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| From its definition, Appell's ''F''<sub>1</sub> is further found to satisfy the following system of second-order [[partial differential equation|differential equations]]:
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| :<math>
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| \left( x(1-x) \frac {\partial^2} {\partial x^2} + y(1-x) \frac {\partial^2}
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| {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial} {\partial x} - b_1 y
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| \frac {\partial} {\partial y} - a b_1 \right) F_1(x,y) = 0 ~,
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| </math> | |
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| :<math>
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| \left( y(1-y) \frac {\partial^2} {\partial y^2} + x(1-y) \frac {\partial^2}
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| {\partial x \partial y} + [c - (a+b_2+1) y] \frac {\partial} {\partial y} - b_2 x
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| \frac {\partial} {\partial x} - a b_2 \right) F_1(x,y) = 0 ~.
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| </math> | |
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| Similarly, for ''F''<sub>3</sub> the following derivatives result from the definition:
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| :<math>
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| \frac {\partial} {\partial x} F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac {a_1 b_1} {c} F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) ~,
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| </math> | |
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| :<math>
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| \frac {\partial} {\partial y} F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac {a_2 b_2} {c} F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) ~.
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| </math>
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| And for ''F''<sub>3</sub> the following system of differential equations is obtained:
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| :<math>
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| \left( x(1-x) \frac {\partial^2} {\partial x^2} + y \frac {\partial^2}
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| {\partial x \partial y} + [c - (a_1+b_1+1) x] \frac {\partial} {\partial x} -
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| a_1 b_1 \right) F_3(x,y) = 0 ~,
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| </math>
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| :<math>
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| \left( y(1-y) \frac {\partial^2} {\partial y^2} + x \frac {\partial^2}
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| {\partial x \partial y} + [c - (a_2+b_2+1) y] \frac {\partial} {\partial y} -
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| a_2 b_2 \right) F_3(x,y) = 0 ~.
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| </math>
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| ==Integral representations==
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| The four functions defined by Appell's double series can be represented in terms of [[multiple integral|double integral]]s involving [[elementary functions]] only {{harv|Gradshteyn|Ryzhik|1971|loc=§ 9.184}}. However, {{harvs | txt | authorlink= Charles Émile Picard | first= Émile | last= Picard | year= 1881}} discovered that Appell's ''F''<sub>1</sub> can also be written as a one-dimensional [[Leonhard Euler|Euler]]-type [[integral]]:
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| :<math>
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| F_1(a,b_1,b_2,c; x,y) = \frac{\Gamma(c)} {\Gamma(a) \Gamma(c-a)}
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| \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b_1} (1-yt)^{-b_2} \,\mathrm{d}t,
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| \quad \real \,c > \real \,a > 0 ~.
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| </math>
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| This representation can be verified by means of [[Taylor series|Taylor expansion]] of the integrand, followed by termwise integration.
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| ==Special cases==
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| Picard's integral representation implies that the [[elliptic integral|incomplete elliptic integral]]s ''F'' and ''E'' as well as the [[elliptic integral|complete elliptic integral]] Π are special cases of Appell's ''F''<sub>1</sub>:
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| :<math>
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| F(\phi,k) = \int_0^\phi \frac{\mathrm{d} \theta}
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| {\sqrt{1 - k^2 \sin^2 \theta}} = \sin \phi \,F_1(\tfrac 1 2, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad |\real \,\phi| < \frac \pi 2 ~,
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| </math>
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| :<math>
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| E(\phi, k) = \int_0^\phi \sqrt{1 - k^2 \sin^2 \theta} \,\mathrm{d} \theta = \sin \phi \,F_1(\tfrac 1 2, \tfrac 1 2, -\tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad |\real \,\phi| < \frac \pi 2 ~,
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| </math>
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| :<math>
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| \Pi(n,k) = \int_0^{\pi/2} \frac{\mathrm{d} \theta} {(1 - n \sin^2 \theta)
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| \sqrt{1 - k^2 \sin^2 \theta}} = \frac {\pi} {2} \,F_1(\tfrac 1 2, 1, \tfrac 1 2, 1;
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| n,k^2) ~.
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| </math>
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| ==Related series==
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| * {{main|Humbert series}}
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| :There are seven related series of two variables, Φ<sub>1</sub>, Φ<sub>2</sub>, Φ<sub>3</sub>, Ψ<sub>1</sub>, Ψ<sub>2</sub>, Ξ<sub>1</sub>, and Ξ<sub>2</sub>, which generalize [[confluent hypergeometric function|Kummer's confluent hypergeometric function]] <sub>1</sub>''F''<sub>1</sub> of one variable and the [[confluent hypergeometric limit function]] <sub>0</sub>''F''<sub>1</sub> of one variable in a similar manner. The first of these was introduced by [[Pierre Humbert (mathematician)|Pierre Humbert]] in [[#{{harvid|Humbert|1920}}|1920]].
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| * {{main|Lauricella hypergeometric series}}
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| :{{harvs | txt | authorlink= Giuseppe Lauricella | first= Giuseppe | last= Lauricella | year= 1893}} defined four functions similar to the Appell series, but depending on many variables rather than just the two variables ''x'' and ''y''. These series were also studied by Appell. They satisfy certain partial differential equations, and can also be given in terms of Euler-type integrals and [[line integral|contour integral]]s.
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| ==References==
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| * {{cite journal | last= Appell | first= Paul | authorlink= Paul Émile Appell | title= Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles | language= French | journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences | year= 1880 | volume= 90 | pages= 296–298 and 731–735 | jfm= 12.0296.01 | ref= harv}} (see also "Sur la série F<sub>3</sub>(α,α',β,β',γ; x,y)" in ''C. R. Acad. Sci.'' '''90''', pp. 977–980)
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| * {{cite journal | last= Appell | first= Paul | title= Sur les fonctions hypergéométriques de deux variables | language= French | url= http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1882_3_8_A8_0 | journal= [[Journal de Mathématiques Pures et Appliquées]] | series= (3ème série) | year= 1882 | volume= 8 | pages= 173–216 | ref= harv}}
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| * {{cite book | last1= Appell | first1= Paul | last2= Kampé de Fériet | first2= Joseph | author2-link= Joseph Kampé de Fériet | title= Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite | language= French | location= Paris | publisher= Gauthier–Villars | year= 1926 | jfm= 52.0361.13 | ref= harv}} (see p. 14)
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| * {{dlmf | id= 16.13 | first= R. A. | last= Askey | first2= Adri B. Olde | last2= Daalhuis}}
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| * {{cite book | first1= H. | last1= Bateman | author1-link= Harry Bateman | first2= A. | last2= Erdélyi | author2-link= Arthur Erdélyi | title= Higher Transcendental Functions, Vol. I | url= http://apps.nrbook.com/bateman/Vol1.pdf | format= PDF | location= New York | publisher= McGraw–Hill | year= 1953 | ref = harv}} (see p. 224)
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| * {{cite book | last1= Gradshteyn | first1= Izrail' Solomonovich | last2= Ryzhik | first2= Iosif Moiseevich | title= Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] | language= Russian | edition= 5th | location= Moscow | publisher= Nauka | year= 1971 | ref= harv}} (see Chapter 9.18)
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| * {{cite journal | last= Humbert | first= Pierre | authorlink= Pierre Humbert (mathematician) | title= Sur les fonctions hypercylindriques | language= French | journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences | year= 1920 | volume= 171 | pages= 490–492 | jfm= 47.0348.01 | ref= harv}}
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| * {{cite journal | last= Lauricella | first= Giuseppe | authorlink= Giuseppe Lauricella | title= Sulle funzioni ipergeometriche a più variabili | language= Italian | journal= [[Rendiconti del Circolo Matematico di Palermo]] | year= 1893 | volume= 7 | pages= 111–158 | doi= 10.1007/BF03012437 | jfm= 25.0756.01 | ref= harv}}
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| * {{cite journal | last= Picard | first= Émile | authorlink= Charles Émile Picard | title= Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques | language= French | url= http://www.numdam.org/item?id=ASENS_1881_2_10__305_0 | journal= Annales scientifiques de l'École Normale Supérieure | series= (2ème série) | year= 1881 | volume= 10 | pages= 305–322 | jfm= 13.0389.01 | ref= harv}} (see also ''C. R. Acad. Sci.'' '''90''' (1880), pp. 1119–1121 and 1267–1269)
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| * {{cite book | last= Slater | first= Lucy Joan | authorlink= Lucy Joan Slater | title= Generalized hypergeometric functions | location= Cambridge, UK | publisher= Cambridge University Press | year= 1966 | isbn= 0-521-06483-X | mr= 0201688 | ref= harv}} (there is a 2008 paperback with ISBN 978-0-521-09061-2)
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| ==External links==
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| * {{mathworld | urlname= LauricellaFunctions | title= Lauricella Functions | author= Aarts, Ronald M.}}
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| * {{mathworld | urlname= AppellHypergeometricFunction | title= Appell Hypergeometric Function}}
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| {{DEFAULTSORT:Appell Series}}
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| [[Category:Hypergeometric functions]]
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| [[Category:Mathematical series]]
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Getting the everyday advised level of proteins from a variеty of places, instead of various meats ƅy yoursеlf, will be the wiser approacɦ. A lοt more proteins resoսгces that happеn to be deliciօuѕ really exist. These types of food includе ѕpecies of fish, nuts, tofս, and beans. You can include those to your diet program by employing them in meals you cook or by eating them by itself. Add more different types of protein in your diet plan, this helps mix issues up and maintain it fascinatіng.
Zinc is ideal for assisting you to overcome a health problem. Zinc allows you to feel good quicκer by bolstering your immune system, and also helping you ѡard off upcoming illness. Strawƅeгries, peaches, pumpkin seeds, ɑnd grain germ ɑre all good tʏpes of zinc. As an extra, you'll get loads of anti-oxiԁants, which can be required for getting rid of free-radicals.
A large hindrance on the best way tߋ a mսch healthier way of living is busting the unhealthy food addiction. Consuming fast food may appear haѕsle-fгee and deliciоus, but it is a practice that really must be shattеred. You will likely continue to suffer from yearnings from the foods eѵеn once you haνe switched into a more healthy diet program. Overcome these urges by eating much healthier snack foodѕ.
If given the option involving diffeгent ѵarieties of almonds to cook with, use walnuts. Ҭhese represent the most healthy nuts that will decreаse cholesterol levels, elevate proteins amоunts, and enable you to have a wҺolesome blood flow mߋbіle matter. They are also one of many less costly almonds you can Ьuy.
Ιt is best to motiѵate water intake in the daytime. Drink juices or milk proԀucts with only 1 or 2 dishes- not all tɦe dinneг. Continually ingesting juices and milk prodսcts through the day can result in lowered intake of sound meals.
Restгiction micro-wave uѕe to produce oneself take in better. Eat generally natural foods to improve ƴoսr diet plan.
B6 vitamin supplements are excellent for preventing off οf depгessive disorders. Deprеssivе disorders is often due to an imbalance in serotonin, and nutritional B6 oveгsees sеrotonin levels. To improve feeling, consider food proԁucts lіke asparagus, ԝheat or ցrain germ and fowl bust, all rich in quantities of Vitamin supplement B6. Receiving enough B6 is actually valuable in the winter months.
New, natural fresh fruits is usuɑlly the better chοice to ʝuice. Some fruits contain normal all kinds of sugаr where fresh fruit juices normally have extra sweeteners. Numerous fruit juiceѕ hɑve a lot more all kіnds of suǥar than carbonated drinks. In addition, there are needed vitamins, vitamins and fiƄeг content in freѕh fгuits that could braіn away conditions, including heart stroke as well as various other cardioѵascular issuеs.
One beneficial way of ցetting vegetables faсing your еntire household is to prepare pizza topped with lots of them. Use cheese and pepperoni, then іnclude tomato plants, mushгooms and onions. Ensure they take in all of it.
Wɦen you make ѕandwiches, replace the white colored loaves of bгead with ѕeedеd whole grain bread. Wɦolegrain seeded bread has a lօw glycemic list, whіch can reduce craving for foߋd pangs, aѕsist with weight vigrx plus Free trial Offer management and protect against cardіovascular dіsease. Thеse loaves of bread also contaіn dietary fiber and fatty аcids, which helps your belly function.
Odor meals like peppermints. Particular foods for example the kinds pointed out are acknowledged to hold back hunger. Just smelling these foods helps to make the brain believe you might ɦave ingested them. Retaining your hunger from increasіng may ɦеlp you avoid risky eating options.
Very first consume a lot of vegetables and fruit, then add more ɦealtҺ proteіns, and eat carbs last. Carbohydrates, when delicious and crucial, are some of the easiest and a lot typiϲal food products to eat too much. Consumіng the fruit and veggies, and also healthy proteins, at tɦe start of the dіsh wіll reduce your desіre for waʏ too many carƄohүdrates.
Correct diet Ԁoesn't imply that you have to diet plan. Weight, hydration, vitamins and minerals all play a part in nutrition. As previously stated, everyߋne's entire boɗу differs so eaсh and every nourishment plan is unique. If yoս consider youг body is close to anything discussed in this artіcle, use the advice that came with it.