The J Transform

Jleft{ fright} left( s right) equiv int_0^{infty} frac{x^{s-1}e^{-x}}{Gamma(s)}  fleft( x right) dx

Jleft{ fright} left( s right) =frac{1}{Gammaleft( sright) } Mleft{ fleft( x right) e^{-x} right} left( s right)  where M is the Mellin transform.[1]

Jleft{ x^nright} left( s right)=frac{Gamma(s+n)}{Gamma(s)}=left( s+n-1 right)left( s+n-2 right) dots left( s right)
Jleft{left(x-cright)^nright} left( s right)=sum_{k=0}^{n}{{n}choose{k}} left( -c right)^{n-k} left( s+k-1 right)left( s+k-2 right) dotsleft( s right)
Jleft{ 1right} left( s right) =1
Jleft{x^{-n}right} left(sright)=frac{1}{left( s-1 right) left( s-2 right) dots left( s-n right) }
Jleft{ x^asinleft( bx right) right} left( s right) =frac{left(s+a-1right)left(s+a-2right)dotsleft(sright)}{left(1+b^2right)^{frac{s+a}{2}}}sinleft(left(s+aright)arctan bright)
Jleft{ x^acosleft( bx right) right} left( s right) =frac{left(s+a-1right)left(s+a-2right)dotsleft(sright)}{left(1+b^2right)^{frac{s+a}{2}}}cosleft(left(s+aright)arctanleft| b right|right)
Jleft{ expleft( cx right) right} left( s right) =frac{1}{left( 1-c right)^{s } }
Jleft{ lnleft( cx right) right} left( s right) =lnleft( c right)  + PolyGammaleft( s right)
Jleft{ frac{1}{x+c}  right} left( s right) =e^c ExpIntegralEleft( s, cright)
Jleft{ sum_{n=0}^{infty} a_n x^nright} left( s right)=a_0+sum_{n=1}^{infty} a_n left( s+n-1 right) left( s+n-2 right) dots left( s right)
Jleft{sum_{n=0}^{infty}a_nleft(x-cright)^nright}left(sright)=sum_{n=0}^infty left( sum_{j=0}^inftyleft(-cright)^j{{j+n}choose{n}}a_{j+n}right) left(s+n-1right)left(s+n-2right)dotsleft(sright)
Jleft{x^n fright} left( s right) = Jleft{fright} left( s+n right)
Jleft{ f'right} left( s right) = Jleft{fright} left( s right)- Jleft{fright} left( s-1 right)
Jleft{f''right}left(sright)= Jleft{fright} left( s right)-2 Jleft{fright} left( s-1 right)+ Jleft{fright} left( s-2 right)
Jleft{f^{left( n right)} right}left(sright)=sum_{k=0}^{n} left(-1right)^k {nchoose k} Jleft{ f right} left( s-k right)
Jleft{ frac{1+x-e^x}{x^n} right} left( s right)=frac{1}{s-n}  frac{1}{left( s-n+2 right)left( s-n+3 right)dotsleft( s-1 right) }
fleft( x right) =e^x sum_{n=0}^{infty}{frac{left(-x right)^n}{n!}}Jleft{fright}left(-nright)=e^x sum_{n=0}^{infty}{frac{left(-xright)^n}{n!}}Jleft{x^{-n}f right} left(0 right)
fleft( x right) =sum_{n=0}^{infty}{frac{1}{n!}}left( Jleft{fright} right)^{left( n right) }left( 0 right) e^x sum_{k=0}^{infty} left( -kright)^n frac{left( -x right)^k }{k!}
Jleft{ left( x-lambda right) f-xf'right} left( s right)=left( 1-lambda right) Jleft{ f  right} left( s right)
Jleft{ e^{cx}f right} left(s right) =sum_{n=0}^{infty}{frac{c^n}{n!} Jleft{ fright} } left( s+n right)
Jleft{ e^{cx}x^k right} left(s right) =sum_{n=0}^{infty}{frac{c^n}{n!} frac{Gamma(s+n+k)}{Gamma(s+k)}
Jleft{x^{2s}f right} left( s right) = frac{Gamma left( -s right)}{Gamma left( s right)} Jleft{f right} left( -s right)
Jleft{ e^x sum_{k=0}^infty left( -k right)^n frac{left( -x right)^k }{k!}  right} left( s right) = s^n
Jleft{ c chi left( a,b right) left( x right)  right} left( s right) =cfrac{Gammaleft( s,a right) -Gammaleft( s,b right)  }{Gammaleft( sright) } , where chi left( a,b right)left( x right)=1 if aleq x<b, else chi left( a,b right)left( x right)=0.
J^{-1}left{ s^n right} left( x right) =e^x left( 1+sum_{k=1}^infty left( -k right)^n frac{left( -x right)^k }{k!}  right)
J^{-1}left{ s^{n+1} right} left( x right) =sleft( J^{-1}left{ s^{n} right}left( x right) - frac{d}{dx} J^{-1}left{ s^{n} right}left( x right) right)
J^{-1}left{ 1right} left( x right) =1
J^{-1}left{ s^{-n} right} left( x right) =e^xleft( 1+ sum_{k=1}^infty frac{1}{k!} frac{left( -x right)^k }{left( -k right)^n}  right)
J^{-1}left(frac{sin s}{left( 1+{tanleft( 1 right)}^2 right)^{frac{s}{2} }} right) left( x right) = sinleft( x tanleft( 1 right)  right)
Jleft{f g'right} left( s right) = Jleft{f gright} left( s right) -Jleft{f' gright} left( s right) -Jleft{f gright} left( s-1 right)
Jleft{frac{f'}{f}right} left( s right) = Jleft{ln fright} left( s right) -Jleft{ln fright} left( s-1 right)
sum_{s=1}^{infty}Jleft{frac{f'}{f}right} left( s right) = lim_{s to infty}Jleft{ln fright} left( s right) -Jleft{ln fright} left( 0 right)
J^{-1}left{ gleft( s right)  right} left( x right) =e^x left( gleft( 1right) +sum_{k=1}^infty  frac{left( -1 right)^k gleft( -k right) }{k!}x^k   right)
sum_{k=0}^infty  frac{ gleft( k right) }{k!}x^k   =e^{x}J^{-1}left{ gleft( -s right)  right} left( -x right) +gleft(0right)-  gleft( -1right) left| Jleft{ fright} left( s right) -Jleft{ f_nright} left( s right) right| =left| int_0^{infty} frac{x^{s-1}e^{-x}}{Gamma(s)}  left( f-f_n right) left( x right) dx right| le int_0^{infty} frac{x^{s-1}e^{-x}}{Gamma(s)}  sup_{0< x<infty}left| f-f_n right|   dx=sup_{0< x<infty}left| f-f_n right|
so if f_n converges uniformly to f on x>0 , then Jleft{ f_nright} converges uniformly to Jleft{ fright}  on Releft(s right) >0.