The Gauss summation formula

F12 (a,b; c;1) =Γ(c) Γ(c-a-b) Γ(c-a) Γ(c-b)

is a well-known entity used repeatedly on this site. One is led to ask if a similar result holds for multivariable generalizations of the Gauss hypergeometric function. Since this information is not readily available on the web, this presentation helps fill that gap.

The primary reference for these multivariable generalizations is Fonctions Hypergeéomeétriques et Hypersphériques by Appell and Kampé de Fériet, available for download here, although only the first third of the book is relevant to this presentation.

The usual derivation of the formula above relies on a recurrence relation increasing the third hypergeometric parameter by unity, which is then extended recursively to infinity. A much simpler derivation starts with the Euler-type integral representation

F12 (a, b; c; x) =1B(a, c-a) 01 dt ta-1 (1-t) c-a-1 (1-xt )b

which can be easily demonstrated to hold by expanding the binomial containing the independent variable using

(1-x )r =k=0 Γ(r +k) Γ(r) xkk!

followed by recognizing an integral definition of the beta function:

Β(x,y) =01 dt tx-1 (1-t )y-1 =Γ(x) Γ(y) Γ(x+y)

The demonstration proceeds as

1B(a, c-a) 01 dt ta-1 (1-t) c-a-1 (1-xt )b =1B(a, c-a) k=0 Γ(b+k) Γ(b) xkk! 01 dt ta+k-1 (1-t) c-a-1 =k=0 Γ(b+k) Γ(b) B(a+k, c-a) B(a, c-a) xkk! =k=0 Γ(a+k) Γ(a) Γ(b+k) Γ(b) Γ(c) Γ(c+k) xkk!

which as expected is the power series definition of the Gauss hypergeometric function.

For the Gauss summation formula, simply set the independent variable equal to unity and recognize another beta function:

F12 (a,b; c;1) =1B(a, c-a) 01 dt ta-1 (1-t) c-a-b -1 F1 2 (a,b; c,1) =B(a, c-a-b) B(a, c-a) =Γ(c) Γ(c-a-b) Γ(c-a) Γ(c-b)

Et voilà! Très simplement!

While it is preferable to write out gamma functions explicitly in the definition of the Gauss hypergeometric function, for multivariable generalizations this becomes cumbersome. The usual notation for Pochhammer symbols is somewhat confusing, not really telling one how to calculate with it and subject to a certain amount of inconsistency. Instead this presentation with adopt Appell’s notation in the reference above, defining

(a,k) Γ(a +k) Γ(a)

so that the Gauss hypergeometric function will be written

F12 (a,b; c;x) =k=0 (a,k) (b,k) (c,k) xkk!

which is more concise without sacrificing legibility.


Now consider multiplying two Gauss hypergeometric functions:

F12 (a,b; c;x) F12 (a, b; c;y) = k,m=0 (a,k) (a,m) (b,k) (b,m) (c,k) (c,m) xk ym k!m!

In order that the double sum not merely separate into two parts, one can replace pairs of Pochhammer symbols

(a,k) (a,m) (b,k) (b,m) (c,k) (c,m)

with the single symbols

(a,k+m) (b,k+m) (c,k+m)

Given the symmetry of the Gauss hypergeometric function, there are only five ways in which this can be done independently: replacing each pair singly, replacing the upper pair together, replacing one upper pair with the lower pair, and replacing all pairs simultaneously.

The last choice does not produce a new function, as can be seen by rewriting the double sum and applying the binomial theorem:

k,m=0 (a,k+m) (b,k+m) (c,k+m) xk ym k!m! =p=0 (a,p) (b,p) (c,p) k=0 p xk yp-k k!(p -k)! =p=0 (a,p) (b,p) (c,p) (x+y )p p! =F12 (a,b; c;x+y)

The other four replacements do produce new functions, which Appell chose as

F1( a,b,b; c;x,y) = k,m=0 (a,k+m) (b,k) (b,m) (c,k+m) xkym k!m! F2( a,b,b; c,c; x,y) = k,m=0 (a,k+m) (b,k) (b,m) (c,k) (c,m) xkym k!m! F3( a,a, b,b; c;x,y) = k,m=0 (a,k) (a,m) (b,k) (b,m) (c,k+m) xkym k!m! F4( a,b; c,c; x,y) = k,m=0 (a,k+m) (b,k+m) (c,k) (c,m) xkym k!m!

Noting that

(a,k+m) =Γ(a+k +m) Γ(a+k) Γ(a+k) Γ(a) =(a+k,m) (a,k)

and performing the sum over m, these functions can also be written

F1( a,b,b; c;x,y) =k=0 (a,k) (b,k) (c,k) F12 (a+k, b; c+k;y) xkk! F2( a,b,b; c,c; x,y) =k=0 (a,k) (b,k) (c,k) F12 (a+k, b; c;y) xkk! F3( a,a, b,b; c;x,y) =k=0 (a,k) (b,k) (c,k) F12 (a, b; c+k;y) xkk! F4( a,b; c,c; x,y) =k=0 (a,k) (b,k) (c,k) F12 (a+k, b+k; c;y) xkk!

which forms are useful for numerical evaluation of the functions.

Now consider expressions for these functions when the independent variables are proportional. With a multiplier on the second variable, the double sum for for the first Appell function can be rewritten as

F1( a,b,b; c;x,xt) = p=0 (a,p) (c,p) xpp! m=0 p (b,p-m) (b,m) p! (p-m)! tmm!

This function is the simplest to rearrange, but the others will have the same general structure: products, possibly inverse, of Pochhammer symbols multiplying a ratio of factorials. In order to identify known functions from these coefficients, they need to be rewritten so that there is no minus sign on the the summation index m. This can be done with an identity for products of gamma functions:

Γ(1-x) Γ(x) =πsinπx

The ratio of factorials becomes

p! (p-m)! =Γ(1 +p) Γ(1+ p-m) =Γ(p +m) Γ(p) sinπ(p +m) -sinπp p! (p-m)! =(p,m) cosπm =(1 )m (p,m)

and the single Pochhammer symbol becomes

(b, p-m) =Γ(1-1 +b+p-m) Γ(b) (b, p-m) =1Γ(b) Γ(1-b-p +m) πsinπ(1 -b-p+m) (b, p-m) =1(1-b -p,m) 1Γ(b) Γ(1-b -p) πsinπ(1 -b-p+m) (b, p-m) =1(1-b -p,m) Γ(b+p) Γ(b) sinπ(b +p) sinπ(1-b -p+m) (b, p-m) =(b,p) (1-b -p,m) ×1 cosπ( 1+m) =(1 )m (b,p) (1-b -p,m)

The first Appell function is then

F1( a,b,b; c;x,xt) = p=0 (a,p) (b,p) (c,p) xpp! m=0 p (p,m) (b,m) (1-b -p,m) tmm! = p=0 (a,p) (b,p) (c,p) F12 (p, b; 1-b-p; t) xpp!

where the Gauss hypergeometric function is a finite polynomial due to the first parameter. The auxiliary variable t helps one to recognized this functional form.

In a similar fashion one can easily show

F2( a,b,b; c,c; x,xt) = p=0 (a,p) (b,p) (c,p) xpp! m=0 p (p,m) (b,m) (1-c -p,m) (1-b -p,m) (c, m) (t)m m! = p=0 (a,p) (b,p) (c,p) F23 (p, b, 1-c-p; 1-b-p, c; t) xpp!


F3( a,a, b,b; c;x,xt) = p=0 (a,p) (b,p) (c,p) xpp! m=0 p (p,m) (a,m) (b,m) (1-a -p,m) (1-b -p,m) (t)m m! = p=0 (a,p) (b,p) (c,p) F23 (p, a, b; 1-a-p, 1-b-p; t) xpp!


F4( a,b; c,c; x,xt) = p=0 (a,p) (b,p) (c,p) xpp! m=0 p (p,m) (1-c -p,m) (c, m) tmm! = p=0 (a,p) (b,p) (c,p) F12 (p, 1-c-p; c; t) xpp!

where the hypergeometric function coefficients are again finite polynomials due to the common first parameter.

Setting t=1 produces expressions for equal independent variables:

F1( a,b,b; c;x,x) = p=0 (a,p) (b,p) (c,p) F12 (p, b; 1-b-p; 1) xpp! F2( a,b,b; c,c; x,x) = p=0 (a,p) (b,p) (c,p) F23 (p, b, 1-c-p; 1-b-p, c; 1) xpp! F3( a,a, b,b; c;x,x) = p=0 (a,p) (b,p) (c,p) F23 (p, a, b; 1-a-p, 1-b-p; 1) xpp! F4( a,b; c,c; x,x) = p=0 (a,p) (b,p) (c,p) F12 (p, 1-c-p; c; 1) xpp!

The expression for the first Appell function can be simplified further by noting that

F12 (p, b; 1-b-p; 1) =Γ(1-b -p) Γ(1-b -b) Γ(1-b) Γ(1-b -b -p) =(1-b,p) (1-b -b, p) =(b +b,p) (b,p)

where the last step uses a modified form of the single Pochhammer symbol above. The first Appell function for equal independent variables is then

F1( a,b,b; c;x,x) = p=0 (a,p) (b+b ,p) (c,p) xpp! =F12 (a, b+b; c;x)

Unfortunately the other three functions do not appear susceptible to any more simplification. There is no general formula for F23 at negative (or positive) unity, and the coefficient of the fourth function is

F12 (p, 1-c-p; c; 1) =Γ(c) Γ(c +c-1 +2p) Γ(c +p) Γ(c+c -1+p)

where the factor of two on the summation index is a significant complication.

One can now state that at least one Appell hypergeometric function has a summation formula like that for the Gauss hypergeometric function, namely

F1( a,b,b; c;1,1) =F12 (a, b+b; c;1) =Γ(c) Γ(c-a-b -b) Γ(c-a) Γ(c-b -b)

which differs from the Gauss form in containing the sum of two b-parameters. Another simpler way to reach this result uses the Euler-type integral representation

F1( a,b,b; c;x,y) =1B(a, c-a) 01 dt ta-1 (1-t) c-a-1 (1-xt )b (1-yt )b

This representation can be confirmed in exactly the same manner as the one above for the Gauss hypergeometric function:

1B(a, c-a) 01 dt ta-1 (1-t) c-a-1 (1-xt )b (1-yt )b =1B(a, c-a) k,m=0 (b,k) (b,m) xkym k!m! 01 dt ta+k+m -1 (1-t )c-a-1 = k,m=0 (b,k) (b,m) B(a+k+m, c-a) B(a, c-a) xkym k!m! = k,m=0 (a,k+m) (b,k) (b,m) (c,k+m) xkym k!m!

This integral representation differs from that of the Gauss hypergeometric function in having one additional binomial in the integrand, providing the additional Pochhammer symbol needed in the series definition. Clearly this behavior can be extended with additional factors, which will be employed below.

Setting both independent variables in the integrand equal to unity and recognizing another beta function gives

F1( a,b,b; c;1,1) =1B(a, c-a) 01 dt ta-1 (1-t) c-a-b -b-1 F1( a,b,b; c;1,1) =B(a, c-a-b -b) B(a,c -a) =Γ(c) Γ(c-a-b -b) Γ(c-a) Γ(c-b -b)

which is the expected result. The longer route through series expansions was taken to indicate whether such expressions might exist for the other three Appell hypergeometric functions, but this does not appear likely.


Now consider multiplying three Gauss hypergeometric functions, slightly shifting notation to use subscripts to track each function. The resulting triple sum will have coefficients of the form

(a1, k1) (a2, k2) (a3, k3) (b1, k1) (b2, k2) (b3, k3) (c1, k1) (c2, k2) (c3, k3)

In order that the sum not merely separate into three parts, one can replace combinations of Pochhammer symbols with single symbols in a variety of ways. Lauricella chose to replace triples

(a1, k1) (a2, k2) (a3, k3) (b1, k1) (b2, k2) (b3, k3) (c1, k1) (c2, k2) (c3, k3)

with the single symbols

(a,k1 +k2 +k3) (b,k1 +k2 +k3) (c,k1 +k2 +k3)

to define four hypergeometric functions that generalize the Appell functions. This was done in an order slightly different from that of Appell, which can cause some confusion:

FA( a;b1, b2, b3; c1,c2, c3; x1,x2, x3) = k1, k2,k3 =0 (a,k1 +k2 +k3) (b1, k1) (b2, k2) (b3, k3) (c1, k1) (c2, k2) (c3, k3) x1k1 x2k2 x3k3 k1! k2! k3! FB(a1, a2, a3; b1,b2, b3; c; x1,x2, x3) = k1, k2,k3 =0 (a1, k1) (a2, k2) (a3, k3) (b1, k1) (b2, k2) (b3, k3) (c,k1 +k2 +k3) x1k1 x2k2 x3k3 k1! k2! k3! FC( a;b; c1,c2, c3; x1,x2, x3) = k1, k2,k3 =0 (a,k1 +k2 +k3) (b,k1 +k2 +k3) (c1, k1) (c2, k2) (c3, k3) x1k1 x2k2 x3k3 k1! k2! k3! FD( a;b1, b2, b3; c; x1,x2, x3) = k1, k2,k3 =0 (a,k1 +k2 +k3) (b1, k1) (b2, k2) (b3, k3) (c,k1 +k2 +k3) x1k1 x2k2 x3k3 k1! k2! k3!

Given that the analysis of Appell hypergeometric functions led to only one significant and useful simplification, it is reasonable that the same would apply for these Lauricella functions. Rather than working through power series expansions again, for expediency consider an Euler-type integral representation of the fourth Lauricella function:

FD( a; b1,b2, b3; c; x1, x2, x3) =1 B(a, c-a) 01 dt ta-1 (1-t) c-a-1 (1-x1t )b1 (1-x2t )b2 (1-x3t )b3

As noted above, adding one more binomial factor to the integrand will simply produce the additional Pochhammer symbol needed in the series definition, so the representation is valid.

Setting all independent variables in the integrand equal to unity gives

FD( a; b1,b2, b3; c; 1,1,1) =1 B(a, c-a) 01 dt ta-1 (1-t) c-a-b1 -b2-b3 -1 =B(a, c-a-b1 -b2 -b3) B(a, c-a) =Γ(c) Γ(c-a -b1-b2 -b3) Γ(c-a) Γ(c-b1 -b2 -b3)

which contains the sum of three b-parameters. A pattern emerges...

The four Lauricella hypergeometric functions can be extended to any number of independent variables accompanied by the appropriate number of parameters. The superscript on the function indicates the number of variables:

FA (n)( a;b1, , bn; c1,, cn; x1,, xn) = k1, , kn=0 (a,k1 + +kn) (b1, k1) (bn, kn) (c1, k1) (cn, kn) x1k1 xnkn k1! kn! FB (n)( a1,, an; b1,, bn; c; x1,, xn) = k1, , kn=0 (a1, k1) (an, kn) (b1, k1) (bn, kn) (c,k1 + +kn) x1k1 xnkn k1! kn! FC (n)( a;b; c1,, cn; x1,, xn) = k1, , kn=0 (a,k1 + +kn) (b,k1 + +kn) (c1, k1) (cn, kn) x1k1 xnkn k1! kn! FD (n)( a;b1, , bn; c; x1,, xn) = k1, , kn=0 (a,k1 + +kn) (b1, k1) (bn, kn) (c,k1 + +kn) x1k1 xnkn k1! kn!

Again focusing on the fourth function, one can extend the previous integral representation with as many binomial factors under the integrand as necessary:

FD (n)( a;b1 , bn; c; x1,, xn) =1 B(a, c-a) 01 dt ta-1 (1-t) c-a-1 (1-x1t )b1 (1-xnt )bn

Setting all independent variables in the integrand equal to unity gives

FD (n)( a;b1 , bn; c; 1,,1) =1 B(a, c-a) 01 dt ta-1 (1-t) c-a-b1 --bn -1 =B(a, c-a-b1 --bn) B(a, c-a) =Γ(c) Γ(c-a -b1 --bn) Γ(c-a) Γ(c-b1 --bn)

and the pattern is clear: the single parameter b in the Gauss summation formula is replaced by the sum of all the b-parameters. Cool!


Uploaded 2024.11.04 analyticphysics.com