Question Number 221582 by MrGaster last updated on 08/Jun/25
![(1):∫_0 ^u x^(ν−1) (u^2 −x^2 )^(ϱ−1) e^(μx) dx=Σ_(n=0) ^∞ (μ^n /(n!))∫_0 ^u x^(ν+n−1) (u^2 −x^2 )^(ϱ−1) dx =Σ_(n=0) ^∞ (μ^n /(n!))∙(u^(2ϱ+ν+n−2) /2)B(((ν+n)/2),ϱ) =(u^(2ϱ+ν+2) /2)Σ_(n=0) ^∞ (((μν)^n )/(n!)) ((Γ(((ν+n)/2))Γ(ϱ))/(Γ(((ν+n)/2)+ϱ))) =((u^(2ρ+ν−2) Γ(ϱ))/2)[Σ_(k=0) ^∞ (((μν)^(2k) )/((2k)!)) ((Γ(ν/2+k))/(Γ(ν/2+ϱ+k)))+Σ_(k=0) ^∞ (((μν)^(2k+1) )/((2k+1))) ((Γ(((ν+1)/2)+k))/(Γ(((ν−1)/2)+ϱ+k)))] =((u^(2ϱ+ν−2) Γ(ϱ))/2)[((Γ(ν/2))/(Γ(ν/2+ϱ))) _1 F_2 ((ν/2);(1/2),(ν/2)+ϱ;((μ^2 u^2 )/4))+((μνΓ(((ν+1)/2)))/(Γ(((ν+1)/2)+ϱ))) _1 F_2 (((ν+1)/2);(3/2),((ν+1)/2)+ϱ;((μ^2 u^2 )/4))] =(1/( 2))B(ν,ϱ)u^(2ν+2ϱ+2) _1 F_2 (ν;(1/2)+ϱ,((μ^2 u^2 )/4))+(μ/2)B(ν+(1/2),ϱ)u^(2ν+2ϱ−1) _1 F_2 (ν+(1/2);(3/2),ν+ϱ+(1/2);((μ^2 u^2 )/4)) (2):=Σ_(k=0) ^∞ (μ^k /(k!))∫_0 ^1 x^(ν+k−1) (u^2 −x^2 )^(ϱ−1) dx =^(x=ut) u^(ν+2ϱ−1) Σ_(k=0) ^∞ (((μu)^k )/(k!))∫_0 ^1 t^(ν+k−1) (1−t^2 )^(ϱ−1) dt =(u^(ν+2ϱ−1) /2)Σ_(k=0) ^∞ (((μν)^k )/(k!))B(((ν+k)/2),ϱ) =(u^(ν+2ϱ−1) /2)[Σ_(m=0) ^∞ (((μν)^(2m) )/((2m)!))B(ν+m,ϱ)+Σ_(m=0) ^∞ (((μu)^(2m+1) )/((2m+1)))B(ν+m+(1/2),ϱ)] =((B(ν,ϱ))/2)u^(ν+2ϱ−1) Σ_(m=0) ^∞ (((ν)_m )/((ν+ϱ)_m )) (((μν/2)^(2m) )/(m!)) +((μB(ν+(1/2),ϱ))/2)u^(ν+2ϱ) Σ_(m=0) ^∞ (((ν+(1/2))_m )/((ν+ϱ+(1/2))_m )) (((μν/2)^(2m) )/((2m+1)!!)) =(1/( 2))B(ν,ϱ)u^(2ν+2ϱ+2) _1 F_2 (ν;(1/2)+ϱ,((μ^2 u^2 )/4))+(μ/2)B(ν+(1/2),ϱ)u^(2ν+2ϱ−1) _1 F_2 (ν+(1/2);(3/2),ν+ϱ+(1/2);((μ^2 u^2 )/4)) (3):∫_0 ^u x^(ν−1) (u^2 −x^2 )^(ϱ−1) e^(μx) dx=(u^(2ϱ+ν−2) /2)Γ(ϱ)Γ((ν/2))Σ_(k=0) ^∞ (((((μ^2 u^2 )/4))^k )/(k!Γ((ν/2)+k+ϱ)Γ(k+(1/2)))) Γ(k+(1/2))=(((2k−1)!!)/2^k )(√π),Γ((ν/2)+k+ϱ)=Γ((ν/2)+ϱ)((ν/2)+ϱ)_k Σ_(k=0) ^∞ (z^k /(k!((ν/2)+ϱ)_k Γ(k+(1/2))))=(1/( Γ(1/2)))Σ_(k=0) ^∞ (((1)_k z^k )/(((ν/2)+ϱ)_k k!((1/2))_k ))=(1/( (√π))) _1 F_2 (1;(1/2),(ν/2)+ϱ;z) _1 F_2 (1;(1/2),(ν/2)+ϱ;z)=Σ_(k=0) ^∞ (((1)_k z^k )/(((1/2))_k ((ν/2)+ϱ)_k k!)) z=((μ^2 u^2 )/4),(1)_k =k! _1 F_2 (1;(1/2),(ν/2)+ϱ;((μ^2 +ν^2 )/4))=Σ_(k=0) ^∞ (((((μ^2 −u^2 )/4))^k )/(((1/2))_k ((ν/2)+ϱ)_k )) =(1/( 2))B(ν,ϱ)u^(2ν+2ϱ+2) _1 F_2 (ν;(1/2)+ϱ,((μ^2 u^2 )/4))+(μ/2)B(ν+(1/2),ϱ)u^(2ν+2ϱ−1) _1 F_2 (ν+(1/2);(3/2),ν+ϱ+(1/2);((μ^2 u^2 )/4)) (4):∫_0 ^u x^(ν−1) (u^2 −x^2 )^(ϱ−1) e^(μx) dx=∫_0 ^u x^(ν−1) (u^2 −x^2 )^(ϱ−1) e^(μx) dx x=ut⇒dx=udt,t∈(0,1) =∫_0 ^1 (ut)^(ν−1) (u^2 (1−t^2 ))^(ϱ−1) e^(μut) udt=u^(ν−1) u^(2ϱ−2) u∫_0 ^1 t^(ν−1) (1−t^2 )^(ϱ−1) e^(μut) dt =u^(v+2ρ−2) ∫_0 ^1 t^(ν−1) (1−t^2 )^(ϱ−1) e^(μut) dt e^(μνt) =Σ_(k=0) ^∞ (((μut)^k )/(k!)) =u^(ν+2ϱ−2) Σ_(k=0) ^∞ (((μu)^k )/(k!))∫_0 ^1 t^(ν+k−1) (1−t^2 )^(ϱ−1) dt t^2 =s⇒t=s^(1/2) ,dt=(1/2)s^(−1/2) ds ∫_0 ^1 t^(ν+k−1) (1−t^2 )^(ϱ−1) dt=∫_0 ^1 s^(((ν+k)/2)−(1/2)) (1−s)^(ϱ−1) ds=(1/2)B(((ν+k)/2),ϱ) =(1/2)u^(ν+2ϱ−2) Σ_(k=0) ^∞ (((μu)^k )/(k!))B(((ν+k)/2),ϱ) B(a,b)=((Γ(a)Γ(b))/(Γ(a+b))) =(1/2)u^(ν+2ϱ−2) Γ(ϱ)Σ_(k=0) ^∞ (((μu)^k )/(k!)) ((Γ(((ν+k)/2)))/(Γ(((ν+k)/2)+ϱ))) k=2m ∧ k=2m+1 =(1/2)u^(ν+2ϱ−2) Γ(a)[Σ_(m=0) ^∞ (((μu)^(2m) )/((2m)!)) ((Γ((ν/2)+m))/(Γ((ν/2)+m+ϱ)))+Σ_(m=0) ^∞ (((μu)^(2m+1) )/((2m+1)!)) ((Γ(((ν+1)/2)+m))/(Γ(((ν−1)/2)+m+ϱ)))] Γ(a+m)=Γ(a)(a)_m =(1/2)u^(ν+2ϱ−2) Γ(a)[((Γ((ν/2)))/(Γ((ν/2)+ϱ)))Σ_(m=0) ^∞ (((μu)^(2m) )/((2m)!))((ν/2))_m /((ν/2)+ϱ)_m +((Γ(((ν+1)/2)))/(Γ(((ν+1)/2)+ϱ)))Σ_(m=0) ^∞ ((((μu)^(2m+1) )/((2m+1)!)) ((Γ(((ν+1)/2)+m))/(Γ(((ν+1)/2)+m+ϱ)))] (1/((2m)!))=(1/(4^m ((1/2))_m m!)),(1/((2m+1)!))=(1/(2^(2m) ((3/2))_m m!))∙(1/(2m+1))∙(1/(Γ((3/2))))Γ((3/2)) (2m)!=4^m ((1/2))_m m!,(2m+1)(2m)!=(2m+1)^(4m) ((1/2))_m m!=2^(2m+1) ((m!)/((−))) ((3/2))=((Γ(m+(3/2)))/(Γ((3/2))))=(((2m+1)!!)/2^m ) (2m+1)!=(2m+1)!!∙2^m m!=(((2m+1)!)/(2^m m!))∙2^m m!=(2m+1)! Σ_(m=0) ^∞ (((μu)^(2m) )/((2m)!)) ((((ν/n))_m )/(((ν/2)+ϱ)_m ))=Σ_(m=0) ^∞ (((((μ^2 u^2 )/4))^m )/(((1/2))_m m!)) ((((ν/2))_m )/(((1/2))_m )) _1 F_2 ((ν/2);(1/2),(ν/2)+ϱ;((μ^2 u^2 )/4)) Σ_(m=0) ^∞ (((μn)^(2m+1) )/((2m)!)) ((((ν/2))_m )/(((ν/2)+ϱ)))= _1 F_2 ((ν/2);(1/2),(ν/2)+ϱ;((μ^2 u^2 )/4)) Σ_(m=0) ^∞ (((μu)^(2m+1) )/((2m+1)!)) (((((ν+1)/2))_m )/(Γ(((ν+1)/2)+ϱ)_m ))=(μu)Σ_(m=0) ^∞ (((μ^2 u^2 )^m )/((2m+1)!)) (((((ν+1)/2))_m )/((((ν+1)/2)+ϱ)))=μu _1 F_2 (((ν+1)/2);(3/2),((ν+1)/2)+ϱ;((μ^2 u^2 )/4)) B((ν/2),ϱ)=((Γ((ν/2))Γ(ϱ))/(Γ((ν/2)+ϱ))),B(((ν+1)/2),ϱ)=((Γ(((ν+1)/2))Γ(ϱ))/(Γ(((ν+1)/2)+ϱ))) Γ(ϱ)=((Γ((ν/2)))/(Γ((ν/2)+ϱ)))=((Γ((ν/2))Γ(ϱ))/(Γ((ν/2)+ϱ)))=B((ν/2),ϱ)=B((ν/2),ϱ) Γ(ϱ)((Γ((ν/2)))/(Γ((ν/2)+ϱ)))=B((ν/2),ϱ) Γ(ϱ)((Γ(((ν+1)/2)))/(Γ(((ν+1)/2)+ϱ)))=B(((ν+1)/2),ϱ) =(1/2)u^(ν+2ϱ−2) [B((ν/2),ϱ)Σ_(m=0) ^∞ (((μu)^(2m) )/((2m)!)) ((((ν/2))_m )/(((ν/2)+ϱ)_m ))+B(((ν+1)/2),ϱ)Σ_(m=0) ^∞ (((μu)^(2m+1) )/((2m+1)!)) (((((ν+1)/2))_m )/((((ν+1)/2)+ϱ)_m ))] Σ_(m=0) ^∞ (((μu)^(2m) )/((2m)!)) ((((v/2))_m )/(((ν/2)+ϱ)))= _1 F_2 ((ν/2);(1/2),(ν/2)+ϱ;((μ^2 u^2 )/4)) Σ_(m=0) ^∞ (((μu)^(2m+1) )/((2m+1)!)) (((((ν+1)/2))_m )/((((ν+1)/2)+ϱ)_m ))=μu _1 F_2 (((ν+1)/2);(1/2);(ν/2)+ϱ;((μ^2 u^2 )/4)) B((ν/2),ϱ)=((Γ((ν/2))Γ(ϱ))/(Γ((ν/2)+ϱ))),B(((ν+1)/2),ϱ)=((Γ(((ν+1)/2))Γ(ϱ))/(Γ(((ν+1)/2)+ϱ))) (1/2)u^(ν+2ϱ−2) B((ν/2),ϱ) _1 F_2 ((ν/2);(1/2),(ν/2)+ϱ;((μ^2 u^2 )/4))+(1/2)u^(ν+2ϱ−2) ∙μuB(((ν+1)/2),ϱ) _1 F_(2 ) (((ν+1)/2);(3/2),((ν+1)/2)+ϱ;((μ^2 u^2 )/4)) =(1/2)B((ν/2),ϱ)u^(ν+2ϱ−2) _1 F_2 ((ν/2);(1/2),(ν/2)+ϱ;((μ^2 u^2 )/4))+(μ/2)B(((ν+1)/2),ϱ)u^(ν+2ϱ−1) _1 F_2 (((ν+1)/2);(3/2),((ν+1)/2)+ϱ;((μ^2 u^2 )/4)) =(1/( 2))B(ν,ϱ)u^(2ν+2ϱ+2) _1 F_2 (ν;(1/2)+ϱ,((μ^2 u^2 )/4))+(μ/2)B(ν+(1/2),ϱ)u^(2ν+2ϱ−1) _1 F_2 (ν+(1/2);(3/2),ν+ϱ+(1/2);((μ^2 u^2 )/4))](https://www.tinkutara.com/question/Q221582.png)
$$\left(\mathrm{1}\right):\int_{\mathrm{0}} ^{{u}} {x}^{\nu−\mathrm{1}} \left({u}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {e}^{\mu{x}} {dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mu^{{n}} }{{n}!}\int_{\mathrm{0}} ^{{u}} {x}^{\nu+{n}−\mathrm{1}} \left({u}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {dx} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mu^{{n}} }{{n}!}\centerdot\frac{{u}^{\mathrm{2}\varrho+\nu+{n}−\mathrm{2}} }{\mathrm{2}}{B}\left(\frac{\nu+{n}}{\mathrm{2}},\varrho\right) \\ $$$$=\frac{{u}^{\mathrm{2}\varrho+\nu+\mathrm{2}} }{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu\nu\right)^{{n}} }{{n}!}\:\frac{\Gamma\left(\frac{\nu+{n}}{\mathrm{2}}\right)\Gamma\left(\varrho\right)}{\Gamma\left(\frac{\nu+{n}}{\mathrm{2}}+\varrho\right)} \\ $$$$=\frac{{u}^{\mathrm{2}\rho+\nu−\mathrm{2}} \Gamma\left(\varrho\right)}{\mathrm{2}}\left[\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu\nu\right)^{\mathrm{2}{k}} }{\left(\mathrm{2}{k}\right)!}\:\frac{\Gamma\left(\nu/\mathrm{2}+{k}\right)}{\Gamma\left(\nu/\mathrm{2}+\varrho+{k}\right)}+\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu\nu\right)^{\mathrm{2}{k}+\mathrm{1}} }{\left(\mathrm{2}{k}+\mathrm{1}\right)}\:\frac{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+{k}\right)}{\Gamma\left(\frac{\nu−\mathrm{1}}{\mathrm{2}}+\varrho+{k}\right)}\right] \\ $$$$=\frac{{u}^{\mathrm{2}\varrho+\nu−\mathrm{2}} \Gamma\left(\varrho\right)}{\mathrm{2}}\left[\frac{\Gamma\left(\nu/\mathrm{2}\right)}{\Gamma\left(\nu/\mathrm{2}+\varrho\right)}\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{\mu\nu\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)}\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu+\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)\right] \\ $$$$=\frac{\mathrm{1}}{\:\mathrm{2}}{B}\left(\nu,\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho+\mathrm{2}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu;\frac{\mathrm{1}}{\mathrm{2}}+\varrho,\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{\mu}{\mathrm{2}}{B}\left(\nu+\frac{\mathrm{1}}{\mathrm{2}},\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho−\mathrm{1}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\nu+\varrho+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$\left(\mathrm{2}\right):=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mu^{{k}} }{{k}!}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\nu+{k}−\mathrm{1}} \left({u}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {dx} \\ $$$$\overset{{x}={ut}} {=}{u}^{\nu+\mathrm{2}\varrho−\mathrm{1}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{{k}} }{{k}!}\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\nu+{k}−\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {dt} \\ $$$$=\frac{{u}^{\nu+\mathrm{2}\varrho−\mathrm{1}} }{\mathrm{2}}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu\nu\right)^{{k}} }{{k}!}{B}\left(\frac{\nu+{k}}{\mathrm{2}},\varrho\right) \\ $$$$=\frac{{u}^{\nu+\mathrm{2}\varrho−\mathrm{1}} }{\mathrm{2}}\left[\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu\nu\right)^{\mathrm{2}{m}} }{\left(\mathrm{2}{m}\right)!}{B}\left(\nu+{m},\varrho\right)+\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}+\mathrm{1}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)}{B}\left(\nu+{m}+\frac{\mathrm{1}}{\mathrm{2}},\varrho\right)\right] \\ $$$$=\frac{{B}\left(\nu,\varrho\right)}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{1}} \underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\nu\right)_{{m}} }{\left(\nu+\varrho\right)_{{m}} }\:\frac{\left(\mu\nu/\mathrm{2}\right)^{\mathrm{2}{m}} }{{m}!} \\ $$$$+\frac{\mu{B}\left(\nu+\frac{\mathrm{1}}{\mathrm{2}},\varrho\right)}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho} \underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\nu+\frac{\mathrm{1}}{\mathrm{2}}\right)_{{m}} }{\left(\nu+\varrho+\frac{\mathrm{1}}{\mathrm{2}}\right)_{{m}} }\:\frac{\left(\mu\nu/\mathrm{2}\right)^{\mathrm{2}{m}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)!!} \\ $$$$=\frac{\mathrm{1}}{\:\mathrm{2}}{B}\left(\nu,\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho+\mathrm{2}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu;\frac{\mathrm{1}}{\mathrm{2}}+\varrho,\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{\mu}{\mathrm{2}}{B}\left(\nu+\frac{\mathrm{1}}{\mathrm{2}},\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho−\mathrm{1}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\nu+\varrho+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$\left(\mathrm{3}\right):\int_{\mathrm{0}} ^{{u}} {x}^{\nu−\mathrm{1}} \left({u}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {e}^{\mu{x}} {dx}=\frac{{u}^{\mathrm{2}\varrho+\nu−\mathrm{2}} }{\mathrm{2}}\Gamma\left(\varrho\right)\Gamma\left(\frac{\nu}{\mathrm{2}}\right)\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)^{{k}} }{{k}!\Gamma\left(\frac{\nu}{\mathrm{2}}+{k}+\varrho\right)\Gamma\left({k}+\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$$$\Gamma\left({k}+\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\left(\mathrm{2}{k}−\mathrm{1}\right)!!}{\mathrm{2}^{{k}} }\sqrt{\pi},\Gamma\left(\frac{\nu}{\mathrm{2}}+{k}+\varrho\right)=\Gamma\left(\frac{\nu}{\mathrm{2}}+\varrho\right)\left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{k}} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{z}^{{k}} }{{k}!\left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{k}} \Gamma\left({k}+\frac{\mathrm{1}}{\mathrm{2}}\right)}=\frac{\mathrm{1}}{\:\Gamma\left(\mathrm{1}/\mathrm{2}\right)}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{1}\right)_{{k}} {z}^{{k}} }{\left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{k}} {k}!\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{k}} }=\frac{\mathrm{1}}{\:\sqrt{\pi}}\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\mathrm{1};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;{z}\right) \\ $$$$\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\mathrm{1};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;{z}\right)=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{1}\right)_{{k}} {z}^{{k}} }{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{k}} \left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{k}} {k}!} \\ $$$${z}=\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}},\left(\mathrm{1}\right)_{{k}} ={k}! \\ $$$$\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\mathrm{1};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} +\nu^{\mathrm{2}} }{\mathrm{4}}\right)=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\frac{\mu^{\mathrm{2}} −{u}^{\mathrm{2}} }{\mathrm{4}}\right)^{{k}} }{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{k}} \left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{k}} } \\ $$$$=\frac{\mathrm{1}}{\:\mathrm{2}}{B}\left(\nu,\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho+\mathrm{2}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu;\frac{\mathrm{1}}{\mathrm{2}}+\varrho,\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{\mu}{\mathrm{2}}{B}\left(\nu+\frac{\mathrm{1}}{\mathrm{2}},\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho−\mathrm{1}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\nu+\varrho+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$\left(\mathrm{4}\right):\int_{\mathrm{0}} ^{{u}} {x}^{\nu−\mathrm{1}} \left({u}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {e}^{\mu{x}} {dx}=\int_{\mathrm{0}} ^{{u}} {x}^{\nu−\mathrm{1}} \left({u}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {e}^{\mu{x}} {dx} \\ $$$${x}={ut}\Rightarrow{dx}={udt},{t}\in\left(\mathrm{0},\mathrm{1}\right) \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left({ut}\right)^{\nu−\mathrm{1}} \left({u}^{\mathrm{2}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)\right)^{\varrho−\mathrm{1}} {e}^{\mu{ut}} {udt}={u}^{\nu−\mathrm{1}} {u}^{\mathrm{2}\varrho−\mathrm{2}} {u}\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\nu−\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {e}^{\mu{ut}} {dt} \\ $$$$={u}^{{v}+\mathrm{2}\rho−\mathrm{2}} \int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\nu−\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {e}^{\mu{ut}} {dt} \\ $$$${e}^{\mu\nu{t}} =\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{ut}\right)^{{k}} }{{k}!} \\ $$$$={u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{{k}} }{{k}!}\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\nu+{k}−\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {dt} \\ $$$${t}^{\mathrm{2}} ={s}\Rightarrow{t}={s}^{\mathrm{1}/\mathrm{2}} ,{dt}=\frac{\mathrm{1}}{\mathrm{2}}{s}^{−\mathrm{1}/\mathrm{2}} {ds} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\nu+{k}−\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\varrho−\mathrm{1}} {dt}=\int_{\mathrm{0}} ^{\mathrm{1}} {s}^{\frac{\nu+{k}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{1}−{s}\right)^{\varrho−\mathrm{1}} {ds}=\frac{\mathrm{1}}{\mathrm{2}}{B}\left(\frac{\nu+{k}}{\mathrm{2}},\varrho\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{{k}} }{{k}!}{B}\left(\frac{\nu+{k}}{\mathrm{2}},\varrho\right) \\ $$$${B}\left({a},{b}\right)=\frac{\Gamma\left({a}\right)\Gamma\left({b}\right)}{\Gamma\left({a}+{b}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \Gamma\left(\varrho\right)\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{{k}} }{{k}!}\:\frac{\Gamma\left(\frac{\nu+{k}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\nu+{k}}{\mathrm{2}}+\varrho\right)} \\ $$$${k}=\mathrm{2}{m}\:\wedge\:{k}=\mathrm{2}{m}+\mathrm{1} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \Gamma\left({a}\right)\left[\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}} }{\left(\mathrm{2}{m}\right)!}\:\frac{\Gamma\left(\frac{\nu}{\mathrm{2}}+{m}\right)}{\Gamma\left(\frac{\nu}{\mathrm{2}}+{m}+\varrho\right)}+\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}+\mathrm{1}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)!}\:\frac{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+{m}\right)}{\Gamma\left(\frac{\nu−\mathrm{1}}{\mathrm{2}}+{m}+\varrho\right)}\right] \\ $$$$\Gamma\left({a}+{m}\right)=\Gamma\left({a}\right)\left({a}\right)_{{m}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \Gamma\left({a}\right)\left[\frac{\Gamma\left(\frac{\nu}{\mathrm{2}}\right)}{\Gamma\left(\frac{\nu}{\mathrm{2}}+\varrho\right)}\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}} }{\left(\mathrm{2}{m}\right)!}\left(\frac{\nu}{\mathrm{2}}\right)_{{m}} /\left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{m}} +\frac{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)}\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\left(\mu{u}\right)^{\mathrm{2}{m}+\mathrm{1}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)!}\:\frac{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+{m}\right)}{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+{m}+\varrho\right)}\right]\right. \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{2}{m}\right)!}=\frac{\mathrm{1}}{\mathrm{4}^{{m}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{m}} {m}!},\frac{\mathrm{1}}{\left(\mathrm{2}{m}+\mathrm{1}\right)!}=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}{m}} \left(\frac{\mathrm{3}}{\mathrm{2}}\right)_{{m}} {m}!}\centerdot\frac{\mathrm{1}}{\mathrm{2}{m}+\mathrm{1}}\centerdot\frac{\mathrm{1}}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)}\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{2}{m}\right)!=\mathrm{4}^{{m}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{m}} {m}!,\left(\mathrm{2}{m}+\mathrm{1}\right)\left(\mathrm{2}{m}\right)!=\left(\mathrm{2}{m}+\mathrm{1}\right)^{\mathrm{4}{m}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{m}} {m}!=\mathrm{2}^{\mathrm{2}{m}+\mathrm{1}} \frac{{m}!}{\left(−\right)} \\ $$$$\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\frac{\Gamma\left({m}+\frac{\mathrm{3}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)}=\frac{\left(\mathrm{2}{m}+\mathrm{1}\right)!!}{\mathrm{2}^{{m}} } \\ $$$$\left(\mathrm{2}{m}+\mathrm{1}\right)!=\left(\mathrm{2}{m}+\mathrm{1}\right)!!\centerdot\mathrm{2}^{{m}} {m}!=\frac{\left(\mathrm{2}{m}+\mathrm{1}\right)!}{\mathrm{2}^{{m}} {m}!}\centerdot\mathrm{2}^{{m}} {m}!=\left(\mathrm{2}{m}+\mathrm{1}\right)! \\ $$$$\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}} }{\left(\mathrm{2}{m}\right)!}\:\frac{\left(\frac{\nu}{{n}}\right)_{{m}} }{\left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{m}} }=\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)^{{m}} }{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{m}} {m}!}\:\frac{\left(\frac{\nu}{\mathrm{2}}\right)_{{m}} }{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{m}} }\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{n}\right)^{\mathrm{2}{m}+\mathrm{1}} }{\left(\mathrm{2}{m}\right)!}\:\frac{\left(\frac{\nu}{\mathrm{2}}\right)_{{m}} }{\left(\frac{\nu}{\mathrm{2}}+\varrho\right)}=\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}+\mathrm{1}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)!}\:\frac{\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)_{{m}} }{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)_{{m}} }=\left(\mu{u}\right)\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu^{\mathrm{2}} {u}^{\mathrm{2}} \right)^{{m}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)!}\:\frac{\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)_{{m}} }{\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)}=\mu{u}\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu+\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$${B}\left(\frac{\nu}{\mathrm{2}},\varrho\right)=\frac{\Gamma\left(\frac{\nu}{\mathrm{2}}\right)\Gamma\left(\varrho\right)}{\Gamma\left(\frac{\nu}{\mathrm{2}}+\varrho\right)},{B}\left(\frac{\nu+\mathrm{1}}{\mathrm{2}},\varrho\right)=\frac{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)\Gamma\left(\varrho\right)}{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)} \\ $$$$\Gamma\left(\varrho\right)=\frac{\Gamma\left(\frac{\nu}{\mathrm{2}}\right)}{\Gamma\left(\frac{\nu}{\mathrm{2}}+\varrho\right)}=\frac{\Gamma\left(\frac{\nu}{\mathrm{2}}\right)\Gamma\left(\varrho\right)}{\Gamma\left(\frac{\nu}{\mathrm{2}}+\varrho\right)}={B}\left(\frac{\nu}{\mathrm{2}},\varrho\right)={B}\left(\frac{\nu}{\mathrm{2}},\varrho\right) \\ $$$$\Gamma\left(\varrho\right)\frac{\Gamma\left(\frac{\nu}{\mathrm{2}}\right)}{\Gamma\left(\frac{\nu}{\mathrm{2}}+\varrho\right)}={B}\left(\frac{\nu}{\mathrm{2}},\varrho\right) \\ $$$$\Gamma\left(\varrho\right)\frac{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)}={B}\left(\frac{\nu+\mathrm{1}}{\mathrm{2}},\varrho\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \left[{B}\left(\frac{\nu}{\mathrm{2}},\varrho\right)\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}} }{\left(\mathrm{2}{m}\right)!}\:\frac{\left(\frac{\nu}{\mathrm{2}}\right)_{{m}} }{\left(\frac{\nu}{\mathrm{2}}+\varrho\right)_{{m}} }+{B}\left(\frac{\nu+\mathrm{1}}{\mathrm{2}},\varrho\right)\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}+\mathrm{1}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)!}\:\frac{\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)_{{m}} }{\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)_{{m}} }\right] \\ $$$$\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}} }{\left(\mathrm{2}{m}\right)!}\:\frac{\left(\frac{{v}}{\mathrm{2}}\right)_{{m}} }{\left(\frac{\nu}{\mathrm{2}}+\varrho\right)}=\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mu{u}\right)^{\mathrm{2}{m}+\mathrm{1}} }{\left(\mathrm{2}{m}+\mathrm{1}\right)!}\:\frac{\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)_{{m}} }{\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)_{{m}} }=\mu{u}\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu+\mathrm{1}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}};\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$${B}\left(\frac{\nu}{\mathrm{2}},\varrho\right)=\frac{\Gamma\left(\frac{\nu}{\mathrm{2}}\right)\Gamma\left(\varrho\right)}{\Gamma\left(\frac{\nu}{\mathrm{2}}+\varrho\right)},{B}\left(\frac{\nu+\mathrm{1}}{\mathrm{2}},\varrho\right)=\frac{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}\right)\Gamma\left(\varrho\right)}{\Gamma\left(\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho\right)} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} {B}\left(\frac{\nu}{\mathrm{2}},\varrho\right)\:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{\mathrm{1}}{\mathrm{2}}{u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \centerdot\mu{uB}\left(\frac{\nu+\mathrm{1}}{\mathrm{2}},\varrho\right)\:_{\mathrm{1}} {F}_{\mathrm{2}\:} \left(\frac{\nu+\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{B}\left(\frac{\nu}{\mathrm{2}},\varrho\right){u}^{\nu+\mathrm{2}\varrho−\mathrm{2}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}},\frac{\nu}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{\mu}{\mathrm{2}}{B}\left(\frac{\nu+\mathrm{1}}{\mathrm{2}},\varrho\right){u}^{\nu+\mathrm{2}\varrho−\mathrm{1}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\frac{\nu+\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\frac{\nu+\mathrm{1}}{\mathrm{2}}+\varrho;\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$=\frac{\mathrm{1}}{\:\mathrm{2}}{B}\left(\nu,\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho+\mathrm{2}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu;\frac{\mathrm{1}}{\mathrm{2}}+\varrho,\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{\mu}{\mathrm{2}}{B}\left(\nu+\frac{\mathrm{1}}{\mathrm{2}},\varrho\right){u}^{\mathrm{2}\nu+\mathrm{2}\varrho−\mathrm{1}} \:_{\mathrm{1}} {F}_{\mathrm{2}} \left(\nu+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}},\nu+\varrho+\frac{\mathrm{1}}{\mathrm{2}};\frac{\mu^{\mathrm{2}} {u}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$