Hs[q_,n_,κ_,c_,k0_,k_]:=Sum[(-1)^σ Binomial[κ,σ]k^n Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))Hypergeometric2F1Regularized[(2-q+n)/2,(3-q+n)/2,1+n,-k^2/(σ c - I k0)^2],{σ,0,κ}]

c=0.1;
κ=4;
k0=1;
k=3;
NIntegrate[(k0 x)^(-2) Exp[I k0 x] x BesselJ[0, k x] (1-Exp[-c x])^κ,{x,0,∞}]

NLimit[ Hs[q,0,4,0.1,1,3],q->2]

Hs[2.001,0,4,0.1,1,3]

Hs2[q_,n_,κ_,c_,k0_,k_]:=Sum[(-1)^σ Binomial[κ,σ]k^n Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))(
π (k^2/(σ c -I k0)^2)^(-(2-q+n)/2)/Gamma[(3-q+n)/2]/Gamma[1+n-(2-q+n)/2]
  Hypergeometric2F1Regularized[(2-q+n)/2,(2-q-n)/2,1/2,-(σ c-I k0)^2/k^2]
-π (k^2/(σ c -I k0)^2)^(-(3-q+n)/2)/Gamma[(2-q+n)/2]/Gamma[1+n-(3-q+n)/2]
  Hypergeometric2F1Regularized[(3-q+n)/2,(3-q-n)/2,3/2,-(σ c-I k0)^2/k^2])
,{σ,0,κ}]

Hs2[2.001,0,4,0.1,1,3]

Hs20[q_,n_,κ_,c_,k0_,k_]:=π Sum[(-1)^σ Binomial[κ,σ]k^n Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))(
 (k^2/(σ c -I k0)^2)^(-(2-q+n)/2)/Gamma[(3-q+n)/2]/Gamma[(q+n)/2]
  Hypergeometric2F1Regularized[(2-q+n)/2,(2-q-n)/2,1/2,-(σ c-I k0)^2/k^2]
- (k^2/(σ c -I k0)^2)^(-(3-q+n)/2)/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2]
  Hypergeometric2F1Regularized[(3-q+n)/2,(3-q-n)/2,3/2,-(σ c-I k0)^2/k^2])
,{σ,0,κ}]

Hs20[2.001,0,4,0.1,1,3]

Hs3[q_,n_,κ_,c_,k0_,k_]:=Sum[(-1)^σ Binomial[κ,σ]k^(q-2) Gamma[2-q+n]Gamma[1+n]
/(2^n k0^q (σ c - I k0)^(3/2(2-q+n)))
    Pochhammer[(3-q+n)/2,-(2-q+n)/2]/Pochhammer[1+n,-(2-q+n)]
Hypergeometric2F1[(2-q+n)/2,(2-q-n)/2,1/2,-(σ-I k0)^2/k^2]
,{σ,0,κ}]

Hs3[2.001,0,4,0.1,1,3]

Hs4[q_,n_,κ_,c_,k0_,k_]:=Sum[(-1)^σ Binomial[κ,σ]k^n /Gamma[1+n] Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))Hypergeometric2F1[(2-q+n)/2,(3-q+n)/2,1+n,-k^2/(σ c - I k0)^2],{σ,0,κ}]

Hs4[2.001,2,4,0.1,1,3]

Hs5[q_,n_,κ_,c_,k0_,k_]:=Sum[(-1)^σ Binomial[κ,σ]k^(q-2)  Gamma[2-q+n]/
(2^n k0^q (σ c - I k0)^(3/2(2-q+n)) Gamma[1+n])
Pochhammer[(3-q+n)/2,-(2-q+n)/2]/Pochhammer[1+n,-(2-q+n)/2]
Hypergeometric2F1[(2-q+n)/2,(2-q-n)/2,1/2,-(σ c - I k0)^2/k^2]
,{σ,0,κ}]

Hs5[2.000,2,4,0.1,1,3]

Hs6[q_,n_,κ_,c_,k0_,k_]:=Sum[Sum[(-1)^σ Binomial[κ,σ]k^(-2+q-2s)  Gamma[2-q+n]π/
(2^n k0^q )(
    Pochhammer[(2-q+n)/2,s]Pochhammer[(2-q-n)/2,s]
  / (Gamma[(3-q+n)/2]Gamma[(q+n)/2]Gamma[1/2+s] s!)
-Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]
  / (Gamma[(2-q+n)/2]Gamma[(q+n-1)/2]Gamma[3/2+s] s!)
      (-σ c-I k0)/k

)
,{σ,0,κ}],{s,0,∞}]

Hs6[2.1,0,4,0.1,1,3]

Hs21[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ]k^n Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))(
 (k^2/(σ c -I k0)^2)^(-(2-q+n)/2)/Gamma[(3-q+n)/2]/Gamma[(q+n)/2]
  Hypergeometric2F1Regularized[(2-q+n)/2,(2-q-n)/2,1/2,-(σ c-I k0)^2/k^2]
- (k^2/(σ c -I k0)^2)^(-(3-q+n)/2)/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2]
  Hypergeometric2F1Regularized[(3-q+n)/2,(3-q-n)/2,3/2,-(σ c-I k0)^2/k^2])
,{σ,0,κ}]

Hs21[2.001,0,4,0.1,1,3]

Hs3a[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ]k^n Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))Sum[
 (k^2/(σ c -I k0)^2)^(-(2-q+n)/2)/Gamma[(3-q+n)/2]/Gamma[(q+n)/2]
 Pochhammer[(2-q+n)/2,s]Pochhammer[(2-q-n)/2,s]/Gamma[1/2+s]/s!
	(-(σ c-I k0)^2/k^2)^s
- (k^2/(σ c -I k0)^2)^(-(3-q+n)/2)/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2]
  Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]/Gamma[3/2+s]/s!
	(-(σ c-I k0)^2/k^2)^s,
{s,0,∞}]
,{σ,0,κ}]

Hs3a[2.001,0,4,0.1,1,3]

//otočené pořadí sumy:

Hs3b[q_,n_,κ_,c_,k0_,k_]:= π Sum[ Sum[(-1)^σ Binomial[κ,σ]k^n Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))(
 (k^2/(σ c -I k0)^2)^(-(2-q+n)/2)/Gamma[(3-q+n)/2]/Gamma[(q+n)/2]
 Pochhammer[(2-q+n)/2,s]Pochhammer[(2-q-n)/2,s]/Gamma[1/2+s]/s!
	(-(σ c-I k0)^2/k^2)^s
- (k^2/(σ c -I k0)^2)^(-(3-q+n)/2)/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2]
  Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]/Gamma[3/2+s]/s!
	(-(σ c-I k0)^2/k^2)^s
)
,{σ,0,κ}]
,{s,0,∞}]

Hs3b[2.001,0,4,0.1,1,3]

Hs4a[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ]k^n Gamma[2-q+n]/(2^n k0^q (σ c - I k0)^(2-q+n))Sum[(-1)^s(
 Pochhammer[(2-q+n)/2,s]Pochhammer[(2-q-n)/2,s]
/Gamma[1/2+s]/s!/Gamma[(3-q+n)/2]/Gamma[(q+n)/2]
k^(-2+q-n-2s)(σ c-I k0)^(2-q+n+2s)
- Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]
/Gamma[3/2+s]/s!/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2]
 k^(-3+q-n-2s)(σ c-I k0)^(3-q+n+2s)
),{s,0,∞}]
,{σ,0,κ}]

Hs4a[2.001,0,4,0.1,1,3]

Hs4b[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ] Gamma[2-q+n]/
(2^n k0^q)
Sum[(-1)^s(
 Pochhammer[(2-q+n)/2,s]Pochhammer[(2-q-n)/2,s]
/Gamma[1/2+s]/s!/Gamma[(3-q+n)/2]/Gamma[(q+n)/2]
k^(-2+q-2s)(σ c-I k0)^(2s)
- Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]
/Gamma[3/2+s]/s!/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2]
 k^(-3+q-2s)(σ c-I k0)^(1+2s)
),{s,0,∞}]
,{σ,0,κ}]

Hs4b[2.001,0,4,0.1,1,3]

Hs4c[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ] Gamma[2-q+n]/
(2^n k0^q)
Sum[(-1)^s k^(-2+q-2s)(σ c-I k0)^(2s)(
 (Pochhammer[(2-q+n)/2,s]Pochhammer[(2-q-n)/2,s]
/Gamma[1/2+s]/s!/Gamma[(3-q+n)/2]/Gamma[(q+n)/2])
- (Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]
/Gamma[3/2+s]/s!/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2])
 (σ c-I k0)/k
),{s,0,∞}]
,{σ,0,κ}]

Hs4c[2.001,0,4,0.1,1,3]

Hs4d[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ] Gamma[2-q+n]/
(2^n k0^q)
Sum[(-1)^s k^(-2+q-2s)(σ c-I k0)^(2s)(
 (Pochhammer[(2-q+n)/2,s]Pochhammer[(2-q-n)/2,s]
/Gamma[1/2+s]/s!/Gamma[(3-q+n)/2]/Gamma[(q+n)/2])
- (Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]
/Gamma[3/2+s]/s!/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2])
 (σ c-I k0)/k
),{s,Ceiling[κ/2],∞}]
,{σ,0,κ}]

Hs4d[2.00001,0,4,0.1,1,3]

Sum[Gamma[s+1/2](-1)^s/(s+1/2)/s!x^s,{s,0,∞}]

Gamma[1/2]

Plot[ArcSinh[1/x],{x,0,200}]

HsSq2n0a[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ] Gamma[2-q+n]/
(2^n k0^q)
Sum[(-1)^s k^(-2+q-2s)(σ c-I k0)^(2s)(
- (Pochhammer[(3-q+n)/2,s]Pochhammer[(3-q-n)/2,s]
/Gamma[3/2+s]/s!/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2])
 (σ c-I k0)/k
),{s,Ceiling[κ/2],∞}]
,{σ,0,κ}]

HsSq2n0a[2.00001,0,4,0.1,1,3]

HsSq2n0b[q_,n_,κ_,c_,k0_,k_]:= π Sum[(-1)^σ Binomial[κ,σ] Gamma[2-q+n]/
(k0^2)
Sum[(-1)^s k^(-2s)(σ c-I k0)^(2s)(
- (Pochhammer[1/2,s]Pochhammer[1/2,s]
/Gamma[3/2+s]/s!/Gamma[(2-q+n)/2]/Gamma[(q+n-1)/2])
 (σ c-I k0)/k
),{s,Ceiling[κ/2],∞}]
,{σ,0,κ}]

HsSq2n0b[2.00001,0,4,0.1,1,3]

HsSpecq2n0f[κ_,c_,k0_,k_]:=-Sqrt[π]k0^(-2)k^(-2)Sum[(-1)^σ Binomial[κ,σ]

(σ c-I k0)^2 ArcSinh[((σ c-I k0)/k)], {σ,0,κ}]

HsSpecq2nf[4,0.1,1,3]

HsSq2n0c[κ_,c_,k0_,k_]:=-Sum[(-1)^σ Binomial[κ,σ]1/(2k0^2)Sum[(-1)^s k^(-2s-1)(σ c-I k0)^(2s+1)Gamma[1/2+s]/(Sqrt[π](1/2+s)s!),{s,0,∞}],{σ,0,κ}]

HsSq2n0c[4,0.1,1,3]

HsSq2n0d[κ_,c_,k0_,k_]:=-Sum[(-1)^σ (σ c-I k0)/k Binomial[κ,σ]1/(2k0^2)Sum[(-1)^s ((σ c-I k0)/k)^(2s)Gamma[1/2+s]/(Sqrt[π](1/2+s)s!),{s,0,∞}],{σ,0,κ}]

HsSq2n0d[4,0.1,1,3]

HsSq2n0e[κ_,c_,k0_,k_]:=-Sum[(-1)^σ Binomial[κ,σ]1/(2k0^2)Sum[(-1)^s ((σ c-I k0)/k)^(2s+1)Gamma[1/2+s]/(Sqrt[π](1/2+s)s!),{s,0,∞}],{σ,0,κ}]

HsSq2n0e[4,0.1,1,3]

HsSq2n0f[κ_,c_,k0_,k_]:=-Sum[(-1)^σ Binomial[κ,σ]1/k0^2ArcSinh[(σ c-I k0)/k],{σ,0,κ}]

HsSq2n0f[4,0.1,1,3]

Sum[(-1)^s (x)^(2s+1)Gamma[1/2+s]/(Sqrt[π](1/2+s)s!),{s,0,∞}]

Sum[(-1)^s x^(2s+1)Gamma[1/2+s]/(Sqrt[π](1/2+s)s!),{s,0,∞}]

HsSq2n0f[2,0.1,1,1.5]

NInt[q_,n_,κ_,c_,k0_,k_]:=NIntegrate[(k0 x)^(-q) Exp[I k0 x] x BesselJ[n, k x] (1-Exp[-c x])^κ,{x,0,∞}]

NInt[2,0,2,0.1,1,1.5]

Small k, q +n <= 2

HsShortPoly2[q_,n_,κ_,c_,k0_,k_]:=2^(1-q)/Sqrt[Pi]Sum[
(-1)^σ Binomial[κ,σ]k^n/(k0^q (σ c-I k0)^(2-q+n))Gamma[1+n]/Gamma[(2-q+n)/2]/Gamma[(3-q+n)/2]
Sum[(-1)^s Binomial[(2-q+n)/2,s]Pochhammer[(3-q+n)/2,s]/Pochhammer[1+n,s]
(-k^2/(σ c-I k0)^2)^s,{s,0,(2-q-n)/2}]
,{σ,0,κ}]

NInt[4,0,4,0.1,1,0.3]

HsShortPoly2[4,0,4,0.1,1,1.5]

HsShort[q_,n_,κ_,c_,k0_,k_]:=2^(1-q)/Sqrt[Pi]Sum[
(-1)^σ Binomial[κ,σ]k^n/(k0^q (σ c-I k0)^(2-q+n))
Sum[Gamma[(2-q+n)/2+s] Gamma[(3-q+n)/2+s]/Gamma[1+n+s]/s!
(-k^2/(σ c-I k0)^2)^s,{s,0,∞}]
,{σ,0,κ}]

HsShort[2.001,0,4,0.8,1,0.3]

NInt[2.001,0,4,0.8,1,0.3]