2.4.5 Special functions

Several orthogonal polynomials and special functions are implemented, using both PARI [GP] and Maxima [Max]. These are documented in the appropriate sections (``Orthogonal polynomials'' and ``Special functions'', respectively) of the Sage reference manual.

sage: x = polygen(QQ, 'x')
sage: chebyshev_U(2,x)
4*x^2 - 1
sage: bessel_I(1,1,"pari",250)
0.56515910399248502720769602760986330732889962162109200948029448947925564096
sage: bessel_I(1,1)
0.565159103992485
sage: bessel_I(2,1.1,"maxima")  # last few digits are random
0.16708949925104899

At this point, Sage has only wrapped these functions for numerical use. For symbolic use, please use the Maxima interface directly, as in the following example:

sage: maxima.eval("f:bessel_y(v, w)")
'?%bessel_y(v,w)'
sage: maxima.eval("diff(f,w)")
'(?%bessel_y(v-1,w)-?%bessel_y(v+1,w))/2'

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