Annoying Summation

[vilidice]

I've been working on a problem for a while now (not homework related), and I've come up against a sum that I can't compute; I've also tried Wolfram Alpha, and it can't process it either (at least not in the amount of time that's allocated). I've tried to find a book that has the sum in our school library but I've come up empty there as well, so I'm hoping that someone here has the experience, or simply outside perspective, to figure this out:

SUM[k=0,INF]([(n+q+1)_k]*Digamma[n+q+1+k]/[(n+2)_k])*(a/t)^k

where (n+q+1)_k denotes the Pochhammer symbol for n+q+1 (Gamma(n+q+1+k)/Gamma(n+q+1)), and Digamma denotes the Digamma Function (http://en.wikipedia.org/wiki/Digamma_function).

The expression arose from running into d/dq(HyperGeometric[1,n+q+1;n+2;a/t]) in a calculation (and resulted in that sum after a small bit of simplification).

Thanks for taking a look in advance.

[Sagekilla]

I'm not 100% sure what you're trying to do (your notation is very hard to read), but I tried the following in Mathematica:

Sum[(Pochhammer[n + q + 1, k] PolyGamma[n + q + 1 + k]) / Pochhammer[n + 2, k] (a/t)^k, {k, 0, \[Infinity]}]

And I received the following expression:

Spoiler:

-(EulerGamma*Hypergeometric2F1[1, 1 + n + q, 2 + n, a/t]) +
(18*a*EulerGamma*t + 24*a*EulerGamma*n*t + 6*a*EulerGamma*n^2*t + 18*a*EulerGamma*q*t + 6*a*EulerGamma*n*q*t +
36*EulerGamma*t^2 + 30*EulerGamma*n*t^2 + 6*EulerGamma*n^2*t^2 -
12*a^2*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] + 12*a^2*EulerGamma*Hypergeometric2F1[1, 3 + n + q, 4 + n,
a/t] - 12*a^2*n*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] +
18*a^2*EulerGamma*n*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] +
6*a^2*EulerGamma*n^2*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] -
12*a^2*q*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] + 18*a^2*EulerGamma*q*
Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] + 12*a^2*EulerGamma*n*q*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] +
6*a^2*EulerGamma*q^2*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t] + 36*t^2*PolyGamma[0, 1 + n + q] +
30*n*t^2*PolyGamma[0, 1 + n + q] + 6*n^2*t^2*PolyGamma[0, 1 + n + q] + 18*a*t*PolyGamma[0, 2 + n + q] +
24*a*n*t*PolyGamma[0, 2 + n + q] + 6*a*n^2*t*PolyGamma[0, 2 + n + q] + 18*a*q*t*PolyGamma[0, 2 + n + q] +
6*a*n*q*t*PolyGamma[0, 2 + n + q] - 24*a^2*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t]*PolyGamma[0, 2 + n + q] -
36*a^2*n*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t]*PolyGamma[0, 2 + n + q] -
12*a^2*n^2*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t]*PolyGamma[0, 2 + n + q] -
36*a^2*q*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t]*PolyGamma[0, 2 + n + q] -
24*a^2*n*q*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t]*PolyGamma[0, 2 + n + q] -
12*a^2*q^2*Hypergeometric2F1[1, 3 + n + q, 4 + n, a/t]*PolyGamma[0, 2 + n + q] -
216*a^2*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 504*a^2*n*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 414*a^2*n^2*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 144*a^2*n^3*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 18*a^2*n^4*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 324*a^2*q*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 486*a^2*n*q*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 234*a^2*n^2*q*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 36*a^2*n^3*q*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 108*a^2*q^2*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 90*a^2*n*q^2*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 18*a^2*n^2*q^2*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 0}, {0, 1}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 144*a^2*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 336*a^2*n*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 276*a^2*n^2*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 96*a^2*n^3*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 12*a^2*n^4*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 216*a^2*q*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 324*a^2*n*q*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 156*a^2*n^2*q*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 24*a^2*n^3*q*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 72*a^2*q^2*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t] - 60*a^2*n*q^2*Gamma[2 + n]*Gamma[3 + n + q]*Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][
{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q}, a/t] - 12*a^2*n^2*q^2*Gamma[2 + n]*Gamma[3 + n + q]*
Derivative[{0, 0, 1}, {0, 0}, 0][HypergeometricPFQRegularized][{1, 3 + n + q, 3 + n + q}, {4 + n, 3 + n + q},
a/t])/(6*(2 + n)*(3 + n)*t^2)

Would help if you told me what were the domains of n, q, a, t. It would simplify the expression considerably.

[vilidice]

Ah, that timed out wolfram when I put it in, probably related to my slow connection.

Here's the sum in more normal notation (I was trying to avoid using the TeX interpreter):

\sum_{k=0}^\infty {(n+q+1)_k \psi_0 (n+q+1+k) \over (n+2)_k}({a \over t})^k = \sum_{k=0}^\infty { {{d\Gamma(n+q+1+k)\over dq} }{\Gamma(n+2) \over \Gamma(n+2+k){\Gamma(n+q+1)}}}({a \over t})^k

The left hand equation is the "simplified" form, while the right hand side is the form using only gamma functions (in case that's easier for people to read).

Where:

(a)_k = {\Gamma(a+k) \over \Gamma(a)}

and

\psi_0 (t)={d \over dt}\ln(\Gamma(t))={{d \over dt}\Gamma(t) \over \Gamma(t)}

Also as for domain restrictions:

n \in N

q \in (0, \infty) \subset R

a \in (0, \infty) \cup (-\infty, 0) \subset R

[legend]

I tried to do some manipulations with the sum, but I didn't get very far. But I noticed the series diverges for q>[something between 1 and 2 ]. If this has any relevance for your problem, I can try to come up with the exact condition and post it here.

[vilidice]

Given the context of the problem it's a bit odd, but not particularly unexpected (or important), so long as within some open ball of 1 the series converges there isn't any particular issue (or to be more precise while q could be in any R, the values I'm most concerned about are in (0,1], the convergence for other values of q can come from slightly altering a definition earlier, and adding two inductions to the overall proof (allowing q to be a function of t such that q(t) is in (0,1], and q has period 1). I hadn't realized this until you mentioned the discontinuity occurring, so the domain for q can be (0,1] with no affect on the final results.

[vilidice]

As an additional note, since the series is a power series with respect to a/t; the series converges for all a/t <= 1, which is to say that a =< t is the region of convergence on the points (a,t). This ends up resulting in something moderately trivial to the overall problem (as a is the lower bound of an integral with upper bound t in another segment of the problem), but confirms the idea that there is some function for which the series converges (which is really something I should have verified before I even started this venture in retrospect).

Also of note, by expanding another term and using the Cauchy Product I was able to get an equally useful summation (which I'm having about the same level of issue solving out):

{\Gamma(n+2) \over \Gamma(n+q+1)} \sum_{k=1}^\infty {\Gamma(n+q+k+1)^3(n+k+3) \over \Gamma(n+k+2)^3\Gamma(k)^2(n+2+k)}({a \over t})^2 \sum_{j=0}^k{\Gamma(k-j)\over \Gamma(k)}\psi(n+q+1+k-j)({t \over a})^j

I don't know if this one's even worse to work with but maybe someone will see something in there that I missed .