find a(k) from binomial series  (for bessel zero order, L transformation Jo(t))
peter.vlasschaert@gmail.com,30/09/2017

info:Differential and Integral calculus vol II
* N.PISKUNOW,Mir Publishers Moscow 
ref : page 295 to 297.

kill(al)$

find a(k)
f=s(x) = 1+a(1)*x+a(2)*x^2+a(3)*x^3+..... +a(n)*x^n+.........
=>  find (1+x)^m = s(x)

use c= value (integer)
want values trough a(k)  , c=k+1 (example c=4)

c:4;

v:1+sum(a(k)*x^k,k,1,c)$
dv:diff(1,x)+sum(diff(a(k)*x^k,x),k,1,c)$

first ODE ,(1+x)*df/dx = m*f ,f(0)=1

p1:(1+x)*dv = m*v;

p2:expand(p1);

p3:ev(p2,x=0);
p4:coeff(p2,x,1);
p5:coeff(p2,x,2);
p6:coeff(p2,x,3);

p7:solve(p4,a(2))[1];
p8:solve(p5,a(3))[1];
p9:solve(p6,a(4))[1];

p10:factor(ev(p7,p3));
p11:factor(ev(p8,p10));
p12:factor(ev(p9,p11));

general recursion equation find from above
(n+1)*a(n+1)+n*a(n)=a(n)*m;

p13:(n+1)*a(n+1)+n*a(n)=a(n)*m;
p14:solve(p13,a(n+1))[1];
p15:ev(p14,n=0);

set a(0)=1 from  'p3'

p16:a(0)=1;
p17:ev(p15,p16);

p18:ev(p14,n=1);
p19:ev(p18,p17);

s(x) infinite serie (binomial series)

p20:s(x)=v;
p21:ev(p20,p3,p10,p11,p12);

infinite serie ,cvg  abs(x),other a(k)'s (use general recursion equation 'p13')

vv:1+sum(a(k)*x^k,k,1,c)+sum(a(k)*x^k,k,5,inf);
p22:ev(vv,p3,p10,p11,p12);

we need this : Laplace Transform for L{Jo(t)},zero bessel function we use ,the above m=-1/2
(1+x)^m = 1+a(1)*x+a(2)*x^2+a(3)*x^3+..... +a(n)*x^n+......................

p23:m=-1/2;

p24:ev(p22,p23)$
p25:ev((1+x)^m=p24,p23);

other example choose :different examples.
m=1/2   =>  find : (1+x)^(1/2)
m=-1/2  =>  x->-x^2 => find : 1/sqrt(1-x^2) => integration 0 to x find:arcsin(x)
m= 1     => find : 1/(1+x) => integration 0 to x :find :ln(1+x) or log(1+x)
m=-1     => x -> - x => find:1/(1-x) => integration 0 to x:find: ln(1-x ) or log(1-x)