No way for general alfa parameter for exact symbolic solution. You get only solution if alfa=0.

dsolve(eval(YOURS-EQUATION, alpha = 0), a(t));

One solution is:

#a(t) = (-4*sqrt(3)*_C1*k*(_C2 - t))^(2/3)/(4*_C1)

Look like solution is: f(x) = 0.

@janhardo 

Is the same expample as yours.

In your qestion is: Zeta(-z) = -2*z!*sin(Pi*z/2)*Zeta(z + 1)/(2*Pi)^(z + 1); in mine answer:

Zeta(-z) +2*z!*sin(Pi*z/2)*Zeta(z + 1)/(2*Pi)^(z + 1)=0

@acer 

(int(1/(x^2*(z - x^a)), x) assuming (0 < a, 0 < z));limit(%, x = x1) - limit(%, x = 1);

in Maple 2021...

(int(1/(x^2*(z - x^a)), x = 1 .. x1, method = _RETURNVERBOSE) assuming (1 < x1, a in real, 0 < z));

give me:

[FAILS = (distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour)]

All methods Fails!!!

Maybe in Maple 2022 can be calculated.

@perr7 

exp(1/x) is only True if m=0.

Try:

rtable([seq(["m" = j, simplify(sum(simplify(eval(m!*x^(m - n)*LaguerreL(m, n - m, x)^2/n!, m = j)), n = 0 .. infinity))], j = 0 .. 2)], subtype = Vector[column]);

Or in Maple 17  should work:

[seq(["m" = j, simplify(sum(simplify(eval(m!*x^(m - n)*LaguerreL(m, n - m, x)^2/n!, m = j)), n = 0 .. infinity))], j = 0 .. 2)];

@mmcdara I'm update my answer.

@perr7 

1.I don't have Maple 17.

2.I used Maple 2021.

3.I changed  sum to add function.

See atached file.

Calculation_ver_2.mw

Update 03.01.2022

I also need a lot of things. Above paper is not free.

Try with maybe works in yours version:

evalf(Int(abs(u(x) - P__2), x = -1 .. 1, method = _CCquad));

?

I would like to see significant improvement of solving analyticaly(symbolicaly):

1.Integrals.

2.Sums.

3.Product.

4.Integral Transfrom and Inverse like Laplace,Fourier,Mellin,Hankel,Hilbert.

5.Sum Transfrom and Inverse like Ztrans.

6. Linear and nonlinear integral equations.

See, for example, integrals the comparison here.

I would like to see significant improvement of solving numerically:

1.A update a pdsolve numerically higher-order multidimensional partial differential equations.

2.A update a pdsolve numerically to solve elliptic PDEs, but now can't handle.

Maybe this helps:

https://www.mapleprimes.com/questions/228656-How-Do-I-Solve-Fractional-Differential

@KIm Gaeun 

Paste Maple code not a picture. What definition is m(s) ?

@nidojan 

@AHSAN 

% - gives the last result generated.