How to force FunctionExpand to work

Questions
1

Hi, I need to obtain explicit formulae for the inverse Laplace transform:
F[k_] = InverseLaplaceTransform[((s/(s + 1))^(k/2))/s, s, t]
for k=1,2,… MATHEMATICA returns an expression involving some Laguerre polynomial, but I have reasons to believe that the result can be expressed by some combinations of exponentials and exponentially scaled Bessel functions BesselI[0,t/2] and BesselI[1,t/2]. Indeed, after many unsuccessfull attempts, I find that
Table[FunctionExpand[F[i]], {i, 1, 25}] produces a desired result. Unfortunately it fails in the case of
i=15, 17, 19, etc. Is there any way to force FunctionExpand[] to provide the relevant expression also in these cases? I also note that (for unclear reasons) InverseLaplaceTransform[] fails to provide the inverse in the case when for example, I enter Sqrt[s/(s+1)]/s, although this is a special case of the above general formula. Lesław

2

HoldForm[LaguerreL[-(k/2), -t] == HypergeometricPFQ[{k/2}, {1}, -t]]

Then:

Table[HypergeometricPFQ[{k/2}, {1}, -t], {k, 15, 30}] // FunctionExpand
 (*some combinations of exponentials and exponentially scaled Bessel functions*)
3

Thanks a lot, this seems to work indeed, but I completely don’t understand why. Where do you know from that LaguerreL[-(k/2), -t] == HypergeometricPFQ[{k/2}, {1}, -t] ?
And why this second form is expanded as I want, whereas the first one is not?
Is there any freely available textbook from which one can learn and understand such tricks?
In my opinion, the main difficulty in using MATHEMATICA is the lack of useful textbooks showing practical techniques. Most of the texts I have seen only copy the various parts of the MATHEMATICA online help but do not explain the actual data processing that takes place. In addition, error messages produced by MATHEMATICA are totally cryptic and do not allow one to understand what to correct in order to eliminate errors. Lesław

4

For First question see here.This site is very helpful.
Or use:

MathematicalFunctionData["LaguerreL","HypergeometricRepresentations"]

(*{Function[{\[FormalNu],\[FormalZ]},Inactivate[LaguerreL[\[FormalNu],\[FormalZ]]\ 
[Equal]HypergeometricPFQ[{-\[FormalNu]},{1},\[FormalZ]]]]}*)

For Second question I don’t know why second form is expanded, whereas the first one is not.
I only ordinary user not a Super Power user form WRI support.

Many years of experience with Mathematica helps a lot with everything.

Regards M.I.