       Help with evaluation of infinite summation

• To: mathgroup at smc.vnet.net
• Subject: [mg14071] Help with evaluation of infinite summation
• From: John Baron <johnb at nova.stanford.edu>
• Date: Sat, 19 Sep 1998 03:42:06 -0400
• Organization: Center for Radar Astronomy, Stanford University, California USA
• Sender: owner-wri-mathgroup at wolfram.com

```I have a power series of the form

f[x_] = Sum[c[m] x^m, {m, 0, Infinity}]

which I would like to evaluate over a fairly large range of x.

The coefficients are defined by a four-term recurrence relation, with

c = 1

c = alpha1 / gam

c = ((2 * gam + a2 * (1 + alpha1)) * c - 2 * (alpha1 + alpha2 +
alpha3)) / (2 * a2 * (1 + gam))

c = (a2 * (4 * (1 + gam) + a2 * (2 + alpha1)) * c -
(2 * a2 * (1 + alpha1 + alpha2 + alpha3) + gam) * c +
(alpha1 + 2 * alpha2)) / (3 * a2^2 * (2 + gam))

c[m_ /; m > 3] := (a2 * (2 * (m - 1) * (m - 2 + gam) +
(m - 1 + alpha1) * a2) * c[m-1] -
((m - 2) * (m - 3 + gam + 2 * a2) +
2 * a2 * (alpha1 + alpha2 + alpha3)) * c[m-2] +
(m - 3 + alpha1 + 2 * alpha2) * c[m-3]) /
(m * a2^2 * (gam + m - 1))

This series is a solution of the differential equation

f''[x] + (gam / x - 1) * f'[x] - (alpha1 / x + 2 * alpha2 / (x - a2) - 2
* alpha2 * alpha3 / (x - a2)^2) * f[x]

which is very similar to the confluent hypergeometric equation, except
for the additional regular singular point at x = a2. The series
converges uniformly for x < a2.

I am interested in calculating f(2*z), 0 < z <~ 100.  f() is also an
implicit function of an integer n, 1 < n <~ z + 4 * z^(1/3), with

alpha = Sqrt[n * (n + 1)]

gam = 2 * (alpha + 1 / 2)

alpha1 = 1 / 2 * (gam + 1 / (4 * z) - 2 * z)

alpha2 = -1 / (16 * z)

alpha3 = -3 / (32 * z)

a2 = 4 * z

Note that for large z, the terms 2 * alpha2 / (x - a2) and 2 * alpha2 *
alpha3 / (x - a2)^2 are small relative to alpha1 / x, so I should be
able to use the confluent hypergeometric function 1F1 here.  However,
I'm having trouble implementing this summation even for small z.  I run
into "Recursion depth exceeded" errors, or if I limit the summation
range to {m, 0, 10}, problems such as "Summand (or its derivative) is
not numerical at point m = 10."

I don't usually use Mathematica, but am using it in this case because
some of these calculations require extended precision arithmetic. I
would appreciate any additional pointers that anyone might be able to
provide.