f[x_, n_, p_]:=
Gamma[n+1]/(Gamma[x+1] Gamma[n-x+1]) *
p^x * (1-p)^(n-x)

Simplify[Sum[x *f[x,n,p],{x, 0, n}]]

pip[p_, alpha_, beta_]:= 1/Beta[alpha, beta] *
p^(alpha-1) * (1-p)^(beta- 1);

Integrate[pip[p,a,b],{p,0,1}]

Simplify[Integrate[ f[x,n,p]*pip[p, a, b],{p, 0,1}]]

(* pin[n_,lam_]:=lam^n /Gamma[n+1] Exp[-lam]; *)
pin[n_,lam_]:=lam^n (1-lam);

Sum[pin[n,l],{n,0,Infinity}]

aa= Sum[ la^n (1-la) Gamma[1+n] Gamma[b+n-x] Gamma[a+x]/(Beta[a,b] *
Gamma[a+b+n] Gamma[1+n-x] Gamma[1+x]),{n,x,Infinity}]

bb= Sum[ n  la^n (1-la) Gamma[1+n] Gamma[b+n-x] Gamma[a+x]/(Beta[a,b] *
Gamma[a+b+n] Gamma[1+n-x] Gamma[1+x]),{n,x,Infinity}]

h=Simplify[bb/aa]

??HypergeometricPFQ

aaa=h/.{x->1000,a->2, b->2, la->0.9}
bbb=h/.{x->1000,a->2, b->2, la->0.95}

Plot[{u, aaa, bbb},{u,0,30}]

aaaa=Table[{l,Sum[ N[ l^n Exp[-l] Gamma[2+n-21] Gamma[2+21]/(Beta[2,2] *Gamma[4+n] Gamma[1+n-21] Gamma[22])],{n,21,50}]}, {l,30,38}]

ListPlot[ aaaa, PlotJoined->True]

aabb= 1-Hypergeometric1F1[2,2+x,lam]/
Hypergeometric1F1[1,2+x,lam]

Plot[ x - aabb/.{lam->5.48}, {x,14.5,15.5}]

Sum[la^n/n!,{n,x+1,Infinity}]

FindRoot[ ( la^(x+1)/x!-Exp[la]+Exp[la]Gamma[x+1,la]/Gamma[x+1]/.{x->15})==0,
{la,10,20}]