2. BINOMIAL SUMS and LIN_DEP

First we verify the Apery-like formulae:

> n:='n':Consider

and 

>

> s := 's':bz:=unapply(evalf(sum(1/(n^s*binomial(2*n,n)),n=1..infinity)),s);

> abz:=unapply(sum((-1)^n/n^s/binomial(2*n,n),n=1..infinity),s);

> Digits:=12; lin_dep([abz(3),Zeta(3)]); lin_dep([bz(4),Zeta(4)]);

BUT...lin_dep([abz(5),Zeta(5)]); Digits:=50; minpoly(abz(5)/Zeta(5),degree=4);

Similar but more complex multiple binomial sumsare uncovered:

Define

> abm:=(alpha,beta)->Sum((-1)^n/n^alpha/binomial(2*n,n)* Sum(1/p^beta,p=1..n-1),n=1..infinity);

*:Digits:=10; s:=[abz(7),abm(3,4),Zeta(7)] :m:=lin_dep(s);

with an EMIPRICAL verification ... of the error evalf(add(m[k]*s[k],k=1..3),50);

> Gen_new:=proc(x) local n,j;
5/2*Sum((-1)^(n+1)/binomial(2*n,n)/n^3/(1-x^4/n^4)
*Product((4*x^4+j^4)/(j^4-x^4),j=1..n-1),n=1..infinity);
end:

Gen_old:=proc(x) local n,j;
Sum(1/n^3/(1-x^4/n^4),n=1..infinity);end:

ID:=proc(x,N) local S;
S:=Gen_old(x)=Gen_new(x);print(`Comparison is`,evalf(S,N));S;end:

> Sum(Zeta(4*n+3)*x^(4*n),n=0..infinity)=Gen_new(x);

> ID(99/100,20);

and a polylog formula for abz(5):r:=(sqrt(5)-1)/2:L:=log(r): n := 'n': k := 'k':v:=[Sum((-1)^(k+1)/k^5/binomial(2*k,k),k=1..infinity), Zeta(5),L^5,L^3*Zeta(2),L^2*Zeta(3), Sum((1/(2*n)^5-L/(2*n)^4)*r^(2*n),n=1..infinity)];

> Digits:=20;lin_dep(v);

>