Hi,
I know that sage has ultraspherical(n,a,x) implemented, however if a
is not a number, ultraspherical(n,a,x) returns the error:
NameError: name 'a' not defined
(even if I write a = var('a')). This, partly, flies in the face of the
fact that the Gegenbauer polynomials are functions of a.
Worse yet, when I try to explicitly define the Gegenbauer polynomial
via:
C(n, a, x) = sum( (2*x)^(n-2*m)*(-1)^(m)*rising_factorial(a, n-m)/
(factorial(m)*factorial(n-2*m)), m, 0, floor(n/2))
(see Higher Transcendental Functions, volume 2, page 175)
sage fails to actually perform the sum. Specifically, it asks for more
constraints (e.g. assume(n+2*a-2>0)). And then, when I impose the
constraints I get;
sage: C(0, a, x).full_simplify()
sum((-1)^m*2^(-2*m)*factorial(a - m - 1)/
(x^(2*m)*factorial(-2*m)*factorial(m)), m, 0, 0)/factorial(a - 1)
sage: C(0,0,x).full_simplify()
0
sage: C(1,a,x).full_simplify()
sum((-1)^m*2^(-2*m + 1)*x^(-2*m + 1)*factorial(a - m)/(factorial(-2*m
+ 1)*factorial(m)), m, 0, 0)/factorial(a - 1)
when I know C(0,a,x) = 1 and C(1,a,x) = 2*a*x.
What's wrong? Why doesn't sage properly simplify this sum?
Thanks,
Steven
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