Special case of hypergeometric function

Trying to figure out if the double sum $$f(x,y)=\sum_{n=0}^{\infty}\sum_{m=n}^{\infty}\dfrac{(a)_n\,(b)_n\,(1)_m}{(a)_m\,(b)_m}\dfrac{x^n\, y^m}{n!\,m!}, \;\; \text{where} \;\; (z)_n=z(z+1)\cdots(z+n-1),$$ is a special case of the generalized hypergeometric function (of two variables). For instance, the Kampé de Fériet function seems to have a similar form, but it looks like Mathematica doesn't support this function yet, alas. It does understand the Appell hypergeometric function though, and $f(x,y)$ seems to have a similar form. Interchanging the inner and outer sums didn't prove useful either.

• Does $(1)_m=m!$ ? If so doesn't $(1)_m/m!=1$ ?
– JimB
Feb 17, 2018 at 17:06
• @JimB: yes, I added it intentionally to make look more like the Kampe function. Of course that doesn’t fix the problem that the inner sum starts from m=n rather than from m=0.
– Alex
Feb 17, 2018 at 18:16
• It seems like a great deal of simplification can take place if summing from $k=m-n=0$ and noting that $(h)_{k+n}=(h)_n (h+n)_k$ and that $(k+n)!=n! (n+1)_k$.
– JimB
Feb 17, 2018 at 18:39

This is more of an extended comment in that the double sum simplifies greatly.

(* Original equation *)
Sum[Pochhammer[a, n] Pochhammer[b, n] Pochhammer[1, m] x^n y^m/
(Pochhammer[a, m] Pochhammer[b, m] n! m!), {n, 0, ∞}, {m, n, ∞}]

(* Change summation from {m,n,∞} to {k,0,∞} with k=m-n *)
f = Sum[Pochhammer[a, n] Pochhammer[b, n] Pochhammer[1, k + n] x^n y^(k + n)/
(Pochhammer[a, k + n] Pochhammer[b, k + n] n! (k + n)!), {n, 0, ∞}, {k, 0, ∞}];

(* Make some replacements *)
f = f //. {Pochhammer[h_, k + n] -> Pochhammer[h, n] Pochhammer[h + n, k],
(k + n)! -> n! Pochhammer[n + 1, k], Pochhammer[1, n] -> n!}

$$\sum _{n=0}^{\infty } \sum _{k=0}^{\infty } \frac{x^n y^{k+n}}{n! (a+n)_k (b+n)_k}$$

Multiply by $1=(1)_k / k!$:

$$\sum _{n=0}^{\infty } \frac{(x y)^n \sum _{k=0}^{\infty } \frac{(1)_k y^k}{k! \left((a+n)_k (b+n)_k\right)}}{n!}$$

and the result is

Sum[HypergeometricPFQ[{1}, {a + n, b + n}, y] (x y)^n/n!, {n, 0, ∞}]

$$\sum _{n=0}^{\infty } \frac{(x y)^n \, _1F_2(1;a+n,b+n;y)}{n!}$$

I don't know if there is further simplification.

• Thank you for your effort. I got the same result. I also swapped the two sums around, and got another answer involving the ${}_2F_2$ function. I couldn't simplify it any further either though. Thanks again anyway.
– Alex
Feb 17, 2018 at 21:56
• You can get your result by just separating the summations: Sum[Sum[Pochhammer[a, n] Pochhammer[b, n] Pochhammer[1, m] x^ n y^m/(Pochhammer[a, m] Pochhammer[b, m] n! m!), {m, n, Infinity}], {n, 0, Infinity}] /. Pochhammer[1, n] -> n! Jul 18, 2018 at 1:05
• @BobHanlon Thanks! That's much simpler.
– JimB
Jul 18, 2018 at 1:10

I'm not sure what is your identification strategy.

z[x_, n_] := Product[x + i, {i, 0, n - 1}]

sum[a_, b_, x_, y_, n_, s_] := Sum[z[a, n] z[b, n] z[1, m] x^n x^m/(z[a, m] z[b, m] n! m!), {m, n, s}]

f[a_, b_, x_, y_, q_, s_] := Sum[sum[a, b, x, y, n, s], {n, 0, q}]

Evaluating the above definitions show that when eg a=b=1,

With[{a = 1., b = 1., s = 13},
f[a, b, x, y, s, s] // Chop
]

$f(x,y)$ seems to converge to the following polynomial

1 + 1. x + 1.25 x^2 + 0.277778 x^3 + 0.529514 x^4 + 0.0573611 x^5 +
0.17021 x^6 + 0.0105575 x^7 + 0.0420872 x^8 + 0.00167832 x^9 +
0.00837987 x^10 + 0.00023243 x^11 + 0.00139363 x^12 + 0.0000284187 x^13
+ 0.000198856 x^14 + 3.10568*10^-6 x^15 +  0.0000248399 x^16 +
3.06576*10^-7 x^17 + 2.7588*10^-6 x^18 +  2.75826*10^-8 x^19 +
2.75801*10^-7 x^20 + 2.27905*10^-9 x^21 + 2.50679*10^-8 x^22 +
1.74067*10^-10 x^23 + 2.08871*10^-9 x^24 +  1.6059*10^-10 x^26

For anything n,m>13 the coefficients get chopped off. Hope this helps.