## 1491 Reputation

8 years, 280 days

## Integral solution....

```Int((1 - sigma*sin(2*Pi*x))^k, x = 0 .. n) = n*hypergeom([1/2 - k/2, -k/2], [1], sigma^2);

f := (k, n, sigma) -> int((1 - sigma*sin(2*Pi*x))^k, x = 0 .. n):
g := (k, n, sigma) -> n*hypergeom([1/2 - 1/2*k, -1/2*k], [1], sigma^2):

f(11/2, 6, -2/13):
evalf(%);

g(11/2, 6, -2/13):
evalf(%);```

Solution only for:  n for Integers !!!

Regards M.I.

## Without use rsolve:...

Solution without use rsolve:

```N := 30;
B := array(0 .. N):
B[0] := 1:
B[1] := -1/2:
for n from 2 to N do
B[n] := -add(binomial(n + 1, k)*B[k], k = 0 .. n - 1)/(n + 1):
end do:
[B[i] \$ (i = 0 .. N)];```

[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, 0, 854513/138, 0, -236364091/2730, 0, 8553103/6, 0, -23749461029/870, 0, 8615841276005/14322]

Analytical solution: for:n >=1 is: -n*Zeta(1 - n).

```[seq(-n*Zeta(1 - n), n = 2 .. N)];

[1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, 0, 854513/138, 0, -236364091/2730, 0, 8553103/6, 0, -23749461029/870, 0, 8615841276005/14322, 0, -7709321041217/510, 0, 2577687858367/6, 0, -26315271553053477373/1919190, 0, 2929993913841559/6, 0, -261082718496449122051/13530, 0, 1520097643918070802691/1806, 0, -27833269579301024235023/690, 0, 596451111593912163277961/282, 0, -5609403368997817686249127547/46410, 0, 495057205241079648212477525/66]```

Why doesn’t this give me any solution ?, because Maple is not a magic box that'll spit out a solution to any problem..All computer algebra systems, including Maple, are limited in their capabilities.

## I doubt there's a closed form for the in...

See attached file.

## I doubt there are simpler form....

Another formula:

`-(-1 + 2*k)*sqrt(1 - csc(omega*T)^2)*(csc(omega*T)^2)^(-1/2 + k)*int(t^(k - 1)/sqrt(1 - t), t = 0 .. 1/csc(omega*T)^2)/sqrt(-1 + csc(omega*T)^2) + 2*sqrt(Pi)*sqrt(1 - csc(omega*T)^2)*(csc(omega*T)^2)^(-1/2 + k)*GAMMA(k)/(sqrt(-1 + csc(omega*T)^2)*GAMMA(-1/2 + k))`

Integral is incomplete beta function ,but Maple dosen't know, solve as hypergeometric function.

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## Maple is not figure it out to give solut...

 > restart;
 > eq1 := 2^(-m/2-n/2)*exp(lambda^2*sigma^2/4)/sqrt(n!)/sqrt(m!*Pi)*int(HermiteH(m,s+lambda*sigma/2)*HermiteH(n,s+lambda*sigma/2)*exp(-s^2), s=-infinity..infinity);;
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## Only simple equation...

We can use Laplace transform and Inverse Laplace transfrom to solve  for: Simple linear-differential  fractional-equations with initial conditions.

differential_equations_with_fractional_order.mw

## Solve delay differential system of equat...

Maple can solve only numerically.Adding random missing values to parameters:

dsolve-delay_sys_example_1.mw

I use Maple version 2023.2,I don't have version 18 !

Regards

## For n=10000...

 > restart;
 > ee := unapply((-1)^n*((-4*n^2 - 16*n - 28)*JacobiP(-1 + n, -1 - 2*n, 2, -1/2) + JacobiP(-2 + n, -2*n, 3, -1/2)*(3 + n)*(-1 + n))*4^n/(48*(1 + n)*n),n):
 >
 > L:=seq(evalb(expand(ee(i))=eee(i+1)), i=1..10000):
 >
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After 30 min computation on my hardware we see that for n =10000 are True.

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## Try....

kernelopts(version);

#`Maple 2023.1, X86 64 WINDOWS, Jul 07 2023, Build ID 1723669`

Try:

```allvalues(value(solll));

If want solve for: y(x) :

```solve(simplify(allvalues(value(solll))), [y(x)]);

```

## Syntax.....

Try:

```with(DynamicSystems):
lambda := 15.4;
f := N1 -> sum(exp(-lambda)*lambda^n/n!, n = 1 .. N1);
Ns := 50;
T := Vector(Ns, t -> t);
A := Vector(Ns, t -> f(t));
DiscretePlot(T, A, style = stair, legend = "stair", color = red, labels = ["time", "signal"]);
plot(f(x), x = 0 .. Ns);```

.

```with(DynamicSystems);
f := N1 -> sum(1/(n^3*sin(n)^2), n = 1 .. N1);
Ns := 400;
T := Vector(Ns, t -> t);
A := Vector(Ns, t -> f(t));
DiscretePlot(T, A, style = stair, legend = "stair", color = red, labels = ["time", "signal"]);```

## Maybe you what:...

From help pages the fractional derivative using the Davison-Essex (D-E) definition:

`diff(f(x),[x\$nu]) = 1/GAMMA(n-nu)*Int((x-t)^(n-nu-1)*diff(f(t),[t\$n]),t = 0 .. x);`

```U1 := t -> (1/2*1/M - 1/4*1/(M*K))*t + 1/2;
eq := Int((t - z)^(ceil(alpha) - alpha - 1)*diff(U1(z), [z \$ ceil(alpha)]), z = 0 .. t)/GAMMA(ceil(alpha) - alpha) + U1(t)/M - U1(t)^2/(M*K) + diff(U1(t), t) - diff(U1(t), t)/epsilon;
(value(eq) assuming (0 < alpha and alpha < 1));
int(%, t);

#t/(2*M) - t/(4*M*K) - ((2/M - 1/(M*K))*t)/(4*epsilon) + (2*M*K*t + t^2*K - 1/2*t^2)/(4*M^2*K) - (1 + (1/M - 1/(2*M*K))*t)^3/(12*M*K*(1/M - 1/(2*M*K))) - (2*K - 1)*t^(2 - alpha)/(4*(-1 + alpha)*M*K*GAMMA(1 - alpha)*(2 - alpha))
```

## With Mathematica:...

Where QPochhammer function.

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