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In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series 2F1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.

Definitions[edit]

The Appell series F1 is defined for |x| < 1, |y| < 1 by the double series

F1(a,b1,b2;c;x,y)=∑m,n=0∞(a)m+n(b1)m(b2)n(c)m+nm!n!xmyn,{\displaystyle 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}~,}

where (q)n{\displaystyle (q)_{n}} is the Pochhammer symbol. For other values of x and y the function F1 can be defined by analytic continuation. It can be shown[1] that

F1(a,b1,b2;c;x,y)=∑r=0∞(a)r(b1)r(b2)r(c−a)r(c+r−1)r(c)2rr!xryr2F1(a+r,b1+r;c+2r;x)2F1(a+r,b2+r;c+2r;y).{\displaystyle F_{1}(a,b_{1},b_{2};c;x,y)=\sum _{r=0}^{\infty }{\frac {(a)_{r}(b_{1})_{r}(b_{2})_{r}(c-a)_{r}}{(c+r-1)_{r}(c)_{2r}r!}}\,x^{r}y^{r}{}_{2}F_{1}\left(a+r,b_{1}+r;c+2r;x\right){}_{2}F_{1}\left(a+r,b_{2}+r;c+2r;y\right)~.}

Similarly, the function F2 is defined for |x| + |y| < 1 by the series

F2(a,b1,b2;c1,c2;x,y)=∑m,n=0∞(a)m+n(b1)m(b2)n(c1)m(c2)nm!n!xmyn{\displaystyle 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}}

and it can be shown[2] that

F2(a,b1,b2;c1,c2;x,y)=∑r=0∞(a)r(b1)r(b2)r(c1)r(c2)rr!xryr2F1(a+r,b1+r;c1+r;x)2F1(a+r,b2+r;c2+r;y).{\displaystyle F_{2}(a,b_{1},b_{2};c_{1},c_{2};x,y)=\sum _{r=0}^{\infty }{\frac {(a)_{r}(b_{1})_{r}(b_{2})_{r}}{(c_{1})_{r}(c_{2})_{r}r!}}\,x^{r}y^{r}{}_{2}F_{1}\left(a+r,b_{1}+r;c_{1}+r;x\right){}_{2}F_{1}\left(a+r,b_{2}+r;c_{2}+r;y\right)~.}

Also the function F3 for |x| < 1, |y| < 1 can be defined by the series

F3(a1,a2,b1,b2;c;x,y)=∑m,n=0∞(a1)m(a2)n(b1)m(b2)n(c)m+nm!n!xmyn,{\displaystyle 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}~,}

and the function F4 for |x|½ + |y|½ < 1 by the series

F4(a,b;c1,c2;x,y)=∑m,n=0∞(a)m+n(b)m+n(c1)m(c2)nm!n!xmyn.{\displaystyle 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}~.}

Recurrence relations[edit]

Like the Gauss hypergeometric series 2F1, the Appell double series entail recurrence relations among contiguous functions. For example, a basic set of such relations for Appell's F1 is given by:

(a−b1−b2)F1(a,b1,b2,c;x,y)−aF1(a+1,b1,b2,c;x,y)+b1F1(a,b1+1,b2,c;x,y)+b2F1(a,b1,b2+1,c;x,y)=0,{\displaystyle (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~,}cF1(a,b1,b2,c;x,y)−(c−a)F1(a,b1,b2,c+1;x,y)−aF1(a+1,b1,b2,c+1;x,y)=0,{\displaystyle 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~,}cF1(a,b1,b2,c;x,y)+c(x−1)F1(a,b1+1,b2,c;x,y)−(c−a)xF1(a,b1+1,b2,c+1;x,y)=0,{\displaystyle 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~,}cF1(a,b1,b2,c;x,y)+c(y−1)F1(a,b1,b2+1,c;x,y)−(c−a)yF1(a,b1,b2+1,c+1;x,y)=0.{\displaystyle 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~.}

Any other relation[3] valid for F1 can be derived from these four.

Similarly, all recurrence relations for Appell's F3 follow from this set of five:

cF3(a1,a2,b1,b2,c;x,y)+(a1+a2−c)F3(a1,a2,b1,b2,c+1;x,y)−a1F3(a1+1,a2,b1,b2,c+1;x,y)−a2F3(a1,a2+1,b1,b2,c+1;x,y)=0,{\displaystyle 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~,}cF3(a1,a2,b1,b2,c;x,y)−cF3(a1+1,a2,b1,b2,c;x,y)+b1xF3(a1+1,a2,b1+1,b2,c+1;x,y)=0,{\displaystyle 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~,}cF3(a1,a2,b1,b2,c;x,y)−cF3(a1,a2+1,b1,b2,c;x,y)+b2yF3(a1,a2+1,b1,b2+1,c+1;x,y)=0,{\displaystyle 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~,}cF3(a1,a2,b1,b2,c;x,y)−cF3(a1,a2,b1+1,b2,c;x,y)+a1xF3(a1+1,a2,b1+1,b2,c+1;x,y)=0,{\displaystyle 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~,}cF3(a1,a2,b1,b2,c;x,y)−cF3(a1,a2,b1,b2+1,c;x,y)+a2yF3(a1,a2+1,b1,b2+1,c+1;x,y)=0.{\displaystyle 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~.}

Derivatives and differential equations[edit]

For Appell's F1, the following derivatives result from the definition by a double series:

∂n∂xnF1(a,b1,b2,c;x,y)=(a)n(b1)n(c)nF1(a+n,b1+n,b2,c+n;x,y){\displaystyle {\frac {\partial ^{n}}{\partial x^{n}}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\left(a\right)_{n}\left(b_{1}\right)_{n}}{\left(c\right)_{n}}}F_{1}(a+n,b_{1}+n,b_{2},c+n;x,y)}∂n∂ynF1(a,b1,b2,c;x,y)=(a)n(b2)n(c)nF1(a+n,b1,b2+n,c+n;x,y){\displaystyle {\frac {\partial ^{n}}{\partial y^{n}}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\left(a\right)_{n}\left(b_{2}\right)_{n}}{\left(c\right)_{n}}}F_{1}(a+n,b_{1},b_{2}+n,c+n;x,y)}

From its definition, Appell's F1 is further found to satisfy the following system of second-order differential equations:

x(1−x)∂2F1(x,y)∂x2+y(1−x)∂2F1(x,y)∂x∂y+[c−(a+b1+1)x]∂F1(x,y)∂x−b1y∂F1(x,y)∂y−ab1F1(x,y)=0{\displaystyle x(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x^{2}}}+y(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{1}+1)x]{\frac {\partial F_{1}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{1}(x,y)}{\partial y}}-ab_{1}F_{1}(x,y)=0}y(1−y)∂2F1(x,y)∂y2+x(1−y)∂2F1(x,y)∂x∂y+[c−(a+b2+1)y]∂F1(x,y)∂y−b2x∂F1(x,y)∂x−ab2F1(x,y)=0{\displaystyle y(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial y^{2}}}+x(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{2}+1)y]{\frac {\partial F_{1}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{1}(x,y)}{\partial x}}-ab_{2}F_{1}(x,y)=0}

A system partial differential equations for F2 is

x(1−x)∂2F2(x,y)∂x2−xy∂2F2(x,y)∂x∂y+[c1−(a+b1+1)x]∂F2(x,y)∂x−b1y∂F2(x,y)∂y−ab1F2(x,y)=0{\displaystyle x(1-x){\frac {\partial ^{2}F_{2}(x,y)}{\partial x^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b_{1}+1)x]{\frac {\partial F_{2}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{2}(x,y)}{\partial y}}-ab_{1}F_{2}(x,y)=0}y(1−y)∂2F2(x,y)∂y2−xy∂2F2(x,y)∂x∂y+[c2−(a+b2+1)y]∂F2(x,y)∂y−b2x∂F2(x,y)∂x−ab2F2(x,y)=0{\displaystyle y(1-y){\frac {\partial ^{2}F_{2}(x,y)}{\partial y^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b_{2}+1)y]{\frac {\partial F_{2}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{2}(x,y)}{\partial x}}-ab_{2}F_{2}(x,y)=0}

The system have solution

F2(x,y)=C1F2(a,b1,b2,c1,c2;x,y)+C2x1−c1F2(a−c1+1,b1−c1+1,b2,2−c1,c2;x,y)+C3y1−c2F2(a−c2+1,b1,b2−c2+1,c1,2−c2;x,y)+C4x1−c1y1−c2F2(a−c1−c2+2,b1−c1+1,b2−c2+1,2−c1,2−c2;x,y){\displaystyle F_{2}(x,y)=C_{1}F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{2}(a-c_{1}+1,b_{1}-c_{1}+1,b_{2},2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{2}(a-c_{2}+1,b_{1},b_{2}-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{2}(a-c_{1}-c_{2}+2,b_{1}-c_{1}+1,b_{2}-c_{2}+1,2-c_{1},2-c_{2};x,y)}

Similarly, for F3 the following derivatives result from the definition:

∂∂xF3(a1,a2,b1,b2,c;x,y)=a1b1cF3(a1+1,a2,b1+1,b2,c+1;x,y){\displaystyle {\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)}∂∂yF3(a1,a2,b1,b2,c;x,y)=a2b2cF3(a1,a2+1,b1,b2+1,c+1;x,y){\displaystyle {\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)}

And for F3 the following system of differential equations is obtained:

x(1−x)∂2F3(x,y)∂x2+y∂2F3(x,y)∂x∂y+[c−(a1+b1+1)x]∂F3(x,y)∂x−a1b1F3(x,y)=0{\displaystyle x(1-x){\frac {\partial ^{2}F_{3}(x,y)}{\partial x^{2}}}+y{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{1}+b_{1}+1)x]{\frac {\partial F_{3}(x,y)}{\partial x}}-a_{1}b_{1}F_{3}(x,y)=0}y(1−y)∂2F3(x,y)∂y2+x∂2F3(x,y)∂x∂y+[c−(a2+b2+1)y]∂F3(x,y)∂y−a2b2F3(x,y)=0{\displaystyle y(1-y){\frac {\partial ^{2}F_{3}(x,y)}{\partial y^{2}}}+x{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{2}+b_{2}+1)y]{\frac {\partial F_{3}(x,y)}{\partial y}}-a_{2}b_{2}F_{3}(x,y)=0}

A system partial differential equations for F4 is

x(1−x)∂2F4(x,y)∂x2−y2∂2F4(x,y)∂y2−2xy∂2F4(x,y)∂x∂y+[c1−(a+b+1)x]∂F4(x,y)∂x−(a+b+1)y∂F4(x,y)∂y−abF4(x,y)=0{\displaystyle x(1-x){\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-y^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b+1)x]{\frac {\partial F_{4}(x,y)}{\partial x}}-(a+b+1)y{\frac {\partial F_{4}(x,y)}{\partial y}}-abF_{4}(x,y)=0}y(1−y)∂2F4(x,y)∂y2−x2∂2F4(x,y)∂x2−2xy∂2F4(x,y)∂x∂y+[c2−(a+b+1)y]∂F4(x,y)∂y−(a+b+1)x∂F4(x,y)∂x−abF4(x,y)=0{\displaystyle y(1-y){\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-x^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b+1)y]{\frac {\partial F_{4}(x,y)}{\partial y}}-(a+b+1)x{\frac {\partial F_{4}(x,y)}{\partial x}}-abF_{4}(x,y)=0}

The system has solution

F4(x,y)=C1F4(a,b,c1,c2;x,y)+C2x1−c1F4(a−c1+1,b−c1+1,2−c1,c2;x,y)+C3y1−c2F4(a−c2+1,b−c2+1,c1,2−c2;x,y)+C4x1−c1y1−c2F4(2+a−c1−c2,2+b−c1−c2,2−c1,2−c2;x,y){\displaystyle F_{4}(x,y)=C_{1}F_{4}(a,b,c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{4}(a-c_{1}+1,b-c_{1}+1,2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{4}(a-c_{2}+1,b-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{4}(2+a-c_{1}-c_{2},2+b-c_{1}-c_{2},2-c_{1},2-c_{2};x,y)}

Integral representations[edit]

The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only (Gradshteyn et al. 2015, §9.184). However, Émile Picard (1881) discovered that Appell's F1 can also be written as a one-dimensional Euler-type integral:

F1(a,b1,b2,c;x,y)=Γ(c)Γ(a)Γ(c−a)∫01ta−1(1−t)c−a−1(1−xt)−b1(1−yt)−b2dt,ℜc>ℜa>0.{\displaystyle F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.}

This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

Special cases[edit]

Picard's integral representation implies that the incomplete elliptic integrals F and E as well as the complete elliptic integral Π are special cases of Appell's F1:

F(ϕ,k)=∫0ϕdθ1−k2sin2⁡θ=sin⁡(ϕ)F1(12,12,12,32;sin2⁡ϕ,k2sin2⁡ϕ),|ℜϕ|<π2,{\displaystyle F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\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 |\Re \,\phi |<{\frac {\pi }{2}}~,}E(ϕ,k)=∫0ϕ1−k2sin2⁡θdθ=sin⁡(ϕ)F1(12,12,−12,32;sin2⁡ϕ,k2sin2⁡ϕ),|ℜϕ|<π2,{\displaystyle 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 |\Re \,\phi |<{\frac {\pi }{2}}~,}Π(n,k)=∫0π/2dθ(1−nsin2⁡θ)1−k2sin2⁡θ=π2F1(12,1,12,1;n,k2).{\displaystyle \Pi (n,k)=\int _{0}^{\pi /2}{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}={\frac {\pi }{2}}\,F_{1}({\tfrac {1}{2}},1,{\tfrac {1}{2}},1;n,k^{2})~.}

Related series[edit]

• There are seven related series of two variables, Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, and Ξ2, which generalize Kummer's confluent hypergeometric function 1F1 of one variable and the confluent hypergeometric limit function 0F1 of one variable in a similar manner. The first of these was introduced by Pierre Humbert in 1920.
• Giuseppe Lauricella (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 contour integrals.

References[edit]

• Appell, Paul (1880). "Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles". Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French). 90: 296–298 and 731–735. JFM 12.0296.01. (see also "Sur la série F3(α,α',β,β',γ; x,y)" in C. R. Acad. Sci. 90, pp. 977–980)
• Appell, Paul (1882). "Sur les fonctions hypergéométriques de deux variables". Journal de Mathématiques Pures et Appliquées. (3ème série) (in French). 8: 173–216. Archived from the original on April 12, 2013.
• Appell, Paul; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13. (see p. 14)
• Askey, R. A.; Olde Daalhuis, A. B. (2010), "Appell series", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• Burchnall, J. L.; Chaundy, T. W. (1940). "Expansions of Appell's double hypergeometric functions". Q. J. Math. First Series. 11: 249–270. doi:10.1093/qmath/os-11.1.249.
• Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I (PDF). New York: McGraw–Hill. (see p. 224)
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "9.18.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.
• Humbert, Pierre (1920). "Sur les fonctions hypercylindriques". Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French). 171: 490–492. JFM 47.0348.01.
• Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili". Rendiconti del Circolo Matematico di Palermo (in Italian). 7: 111–158. doi:10.1007/BF03012437. JFM 25.0756.01. S2CID 122316343.
• Picard, Émile (1881). "Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques". Annales Scientifiques de l'École Normale Supérieure. Série 2 (in French). 10: 305–322. doi:10.24033/asens.203. JFM 13.0389.01. (see also C. R. Acad. Sci. 90 (1880), pp. 1119–1121 and 1267–1269)
• Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. (there is a 2008 paperback with ISBN 978-0-521-09061-2)

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