Hello,
I have tried to write some maxima functions to reduce
the Heun function and the Appell F1 hypergeometric function
of two variables to simpler expressions, similar to what the hgfred
function does with pFq hypergeometric functions.
These two functions are included below as attachments.
They seem to give correct results for the limited tests
I did, but I am unsure that they are able to detect an incorrect
input and that they will not fail to detect some possible
simplifications.
I would like to know if a similar package already exists for Maxima,
and if not whether these functions with appropriate corrections
can be put at the disposition of Maxima users via some repository.
Sincerely,
E. Orignac
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appellf1red(a,b,c,d,x,y):=block([],
/* We reduce the Appell hypergeometric functions of 2 variables to simpler functions of one variable using the identities listed in http://functions.wolfram.com or the original paper of Paul Appell */
if x = 0 then hgfred([a,c],[d],y)
else if y=0 then hgfred([a,b],[d],x)
else if x=1 then hgfred([a,b],[d],1)*hgfred([a,c],[d-b],y)
else if y=1 then hgfred([a,c],[d],1)*hgfred([a,b],[d-c],x)
else if (x=y) then hgfred([a,b+c],[d],x)
else if (x=-y and b=c) then hgfred([(a+1)/2,a/2,b],[(d+1)/2,d/2],x**2)
else if (d=b+c) then (1-y)^(-a)*hgfred([a,b],[d],(x-y)/(1-y))
/* Some identities with elliptic functions are not included due to G&R's
definition not being in accordance with A&S' definition */
/* Identity from Ismail and Pitman, Can. J. Math. 52 p. 961 (2000) */
else if (b=c and d=2*a and x*y=x+y and(abs(x)<1/2 or abs(y)<1/2)) then hgfred([a,b],[a+1/2],x*y/4)
/* Identity from Ira Gessel, quoted in Ismail and Pitman op. cit. */
else if (d=2*a and b=c and b=a+1/2 and abs(x)<1 and abs(y)<1) then 1/2*(1+1/(sqrt(1-y)*sqrt(1-z)))*(2/(sqrt(1-y)+sqrt(1-z)))^(2*a)
else appell_f1([a],[b,c],[d],x,y) );
/* Source hep-ph/9511322 Eqs. (B.1) to (B.6) */
appellf3red(a,a1,b,b1,c,x,y):=block([],
if (x=0) then return(hgfred([a1,b1],[c],y)),
if (y=0) then return(hgfred([a1,b1],[c],y)),
if (a+a1=c) then return(appellf1red(a,b,b1,c,x,y/(y-1))/(1-y)^b1),
return(appell_f3([a,a1],[b,b1],[c],x,y)))$
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/* Source of identities for Heun's function:
G. Valent, math-ph/0512006 */
heun_red(l1,l2,w):=block([rho,r,k2,s,a,b,c,d],
if (not(listp(l1)) or not(listp(l2))) then return(false),
print (l1),
print (l2),
if ((length(l1)#2) or (length(l2)#4)) then return(false),
k2:first(l1),
s:last(l1),
a:first(l2),
b:second(l2),
c:third(l2),
d:fourth(l2),
print (k2),
print (s),
rho:(c+d-1)/2,
if (k2=0) then return(hgfred([rho+sqrt(rho^2+s),rho-sqrt(rho^2+s)],[c],w)),
rho:(c-a-b)/2,
r:rho-sqrt(rho^2-a*b-s),
if (k2=1) then return(hgfred([r+a,r+b],[c],w)),
if ((k2=-1) and (s=0) and ((a+b+1)=(d+2*c))) then return(hgfred([a,b],[c],w^2)),
if ((k2=1/2) and ((a*b)=(-2*s)) and (d=(1+a+b-2*c))) then return(hgfred([a,b],[c],w*(2-w))),
if ((k2=2) and (s=(-a*b)) and (c=d)) then return(hgfred([a,b],[c],4*w*(1-w))),
if ((k2=1/4) and (s=(-a*b/4)) and (c=1/2) and (d=2*(a+b)/3)) then return (hgfred([a,b],[1/2],w*(3-w)^2/4)),
if ((k2=1/2) and (s=(-a*b/2)) and (c=((a+b)/4+1/2)) and (d=(a+b)/2)) then return(hgfred([a,b],[a+b+1/2],4*w*(2-w)*(1-w)^2)),
if ((k=1/4) and (s=-1/8) and (a=1/2) and (b=1/2) and (c=1) and (d=1/2)) then return(sqrt(sqrt(4-w)-sqrt(1-w))*hgfred([1/2,1/2],[1],(2-w*sqrt(4-w)-(2-w)*sqrt(1-w)))),
/* Carlitz solutions */
return(heun_hn([k2,s],[a,b,c,d],w))
)$