I begin with
FullSimplify[Series[HypergeometricPFQ[{-k,m/2-1,(m-1)/2,m/2},{1/2,1,3m/2-k-2},t],{t,0,3}],
Assumptions->k\[Element]Integers\[And]m\[Element]Integers\[And]k>=0\[And]m>0]
and obtain
Now I want to get a formula for the $j$th coefficient. Of course it is easy to guess it but then I should prove it, and also there are more complicated cases. So I try
FullSimplify[SeriesCoefficient[HypergeometricPFQ[{-k,m/2-1,(m-1)/2,m/2},{1/2,1,3m/2-k-2},t],{t,0,j}],
Assumptions->k\[Element]Integers\[And]m\[Element]Integers\[And]k>=0\[And]m>0]
The result (on 11.0.1.0) is zero. I would understand if the engine would reply that it cannot compute it, but why zero??
k>=0assumption makes it zero. Are you sure that's correct? $\endgroup$k>=0I seem to obtain correct result. Which is even more strange since there are definitely nonzero cases with nonnegativek. You can observe this by computing the above series using, say,With[{k=3},...]$\endgroup$FullSimplify[ Piecewise[{{1/(-1 - k)!, j >= 0}}, 0], Assumptions -> k \[Element] Integers \[And] m \[Element] Integers \[And] k >= 0 \[And] m > 0]returns zero. So doesTable[1/(-1 - k)!, {k, 100}]. $\endgroup$kapproaches an integer? However,Limit[(Gamma[j - k] Gamma[-2 - k + (3 m)/2])/(Gamma[-k] Gamma[-2 + j - k + (3 m)/2]), k -> k0, Assumptions -> k0 \[Element] Integers \[And] m \[Element] Integers \[And] m > 0 && j >= 0 && k0 >= 3]returns0even though the limits at specific values ofjandmare generally nonzero. I'd be inclined to call the zero limit a bug, since it's not even generically true. $\endgroup$