Simplifying $sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}$, where $x$ is an integer and $a<1$











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I would like to simplify the following expression,



$$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}$$



where $x$ is an integer and $a<1$.



Is it possible to lose the sum?



An approximation for the sum will be also helpful.










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  • presumably $x$ is an integer?
    – Seth
    Nov 28 at 22:21










  • Yes @Seth thanks.
    – Y.L
    Nov 28 at 22:25










  • you can bring out the $a^x$ factor and also $1/x!$
    – phdmba7of12
    Nov 28 at 22:30










  • Thanks @phdmba7of12 but what with the sum?
    – Y.L
    Nov 28 at 22:39















up vote
0
down vote

favorite












I would like to simplify the following expression,



$$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}$$



where $x$ is an integer and $a<1$.



Is it possible to lose the sum?



An approximation for the sum will be also helpful.










share|cite|improve this question
























  • presumably $x$ is an integer?
    – Seth
    Nov 28 at 22:21










  • Yes @Seth thanks.
    – Y.L
    Nov 28 at 22:25










  • you can bring out the $a^x$ factor and also $1/x!$
    – phdmba7of12
    Nov 28 at 22:30










  • Thanks @phdmba7of12 but what with the sum?
    – Y.L
    Nov 28 at 22:39













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I would like to simplify the following expression,



$$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}$$



where $x$ is an integer and $a<1$.



Is it possible to lose the sum?



An approximation for the sum will be also helpful.










share|cite|improve this question















I would like to simplify the following expression,



$$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}$$



where $x$ is an integer and $a<1$.



Is it possible to lose the sum?



An approximation for the sum will be also helpful.







calculus real-analysis sequences-and-series taylor-expansion






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share|cite|improve this question













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share|cite|improve this question








edited Nov 28 at 23:17

























asked Nov 28 at 22:19









Y.L

597




597












  • presumably $x$ is an integer?
    – Seth
    Nov 28 at 22:21










  • Yes @Seth thanks.
    – Y.L
    Nov 28 at 22:25










  • you can bring out the $a^x$ factor and also $1/x!$
    – phdmba7of12
    Nov 28 at 22:30










  • Thanks @phdmba7of12 but what with the sum?
    – Y.L
    Nov 28 at 22:39


















  • presumably $x$ is an integer?
    – Seth
    Nov 28 at 22:21










  • Yes @Seth thanks.
    – Y.L
    Nov 28 at 22:25










  • you can bring out the $a^x$ factor and also $1/x!$
    – phdmba7of12
    Nov 28 at 22:30










  • Thanks @phdmba7of12 but what with the sum?
    – Y.L
    Nov 28 at 22:39
















presumably $x$ is an integer?
– Seth
Nov 28 at 22:21




presumably $x$ is an integer?
– Seth
Nov 28 at 22:21












Yes @Seth thanks.
– Y.L
Nov 28 at 22:25




Yes @Seth thanks.
– Y.L
Nov 28 at 22:25












you can bring out the $a^x$ factor and also $1/x!$
– phdmba7of12
Nov 28 at 22:30




you can bring out the $a^x$ factor and also $1/x!$
– phdmba7of12
Nov 28 at 22:30












Thanks @phdmba7of12 but what with the sum?
– Y.L
Nov 28 at 22:39




Thanks @phdmba7of12 but what with the sum?
– Y.L
Nov 28 at 22:39










3 Answers
3






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2
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accepted










The sum can be written as
$$
eqalign{
& S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
& = e^{, - ,a} sumlimits_{k = x}^infty {left( {k - x} right){{a^{,k} } over {k!}}} = e^{, - ,a} left( {sumlimits_{k = 0}^infty
{left( {k - x} right){{a^{,k} } over {k!}}} - sumlimits_{k = 0}^{x - 1} {left( {k - x} right){{a^{,k} } over {k!}}} } right) = cr
& = e^{, - ,a} left( {sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} - xsumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
- sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} + xsumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) cr}
$$



Since
$$
left{ matrix{
e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = {{Gamma (x,a)} over {Gamma (x)}} = Q(x,a) hfill cr
sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} = a{d over {da}}sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}} = ae^{,,a} hfill cr
sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} = sumlimits_{k = 1}^{x - 1} {{{a^{,k} } over {left( {k - 1} right)!}}}
= asumlimits_{j = 0}^{x - 2} {{{a^{,j} } over {j!}}} = ae^{,,a} Q(x - 1,a) hfill cr} right.
$$



where $Q(x,a)$ is the Regularized Incomplete Gamma function,

it is easy to conclude.



Another way is
$$
eqalign{
& S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
& = e^{, - ,a} left( {sumlimits_{n = 0}^infty {left( {n + x} right){{a^{,x + n} } over {left( {x + n} right)!}}}
- xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
& = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,x + n - 1} } over {left( {x + n - 1} right)!}}}
- xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
& = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
- xsumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n + 1} } over {left( {left( {x - 1} right) + n + 1} right)!}}} } right) = cr
& = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
- xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
& = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + asumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} }
over {left( {left( {x - 1} right) + n} right)!}}}
- xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
& = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
+ left( {a - x} right)sumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
& = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
+ left( {a - x} right)sumlimits_{k = x}^infty {{{a^{,k} } over {k!}}} } right) = cr
& = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + left( {a - x} right)sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
- left( {a - x} right)sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) = cr
& = e^{, - ,a} {{a^{,x} } over {left( {x - 1} right)!}} + left( {a - x} right)
- left( {a - x} right)e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = cr
& = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right)left( {1 - Q(x,a)} right) = cr
& = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right){{gamma (x,a)} over {Gamma (x)}} cr}
$$

which checks with the previous one.



--- Addendum ---



Concerning your comment and the request for a nice approximation to the sum $S(a,x)$,
it is not much clear what you intend / need to do. In any case consider the following:

- the formula above (1st and 2nd way lead to the same result) is valid for $a$ and $c$
real or even complex ( with some limitations);

- if $x$ is an integer as you say, then for small values of it the expression in the last-but-third line
is easily computable;

- if $x$ is large , we can apply to $Gamma(x)$ and $Q(x,a)$ the known asymptotic expansions
which are
$$
eqalign{
& Gamma (x) = sqrt {,{{2,pi } over x},} left( {{x over e}} right)^{,x} left( {1 + Oleft( {{1 over x}} right)} right)
quad left| {;x, to ,infty ,;;left| {,arg (x),} right|} right. < pi cr
& Q(x,a) approx 1 - {{x^{, - x - 1/2} e^{,x - a} a^{,x} } over {sqrt {2pi } }}left( {1 + {{12a - 1} over {12x}}
+ Oleft( {{1 over {x^{,2} }}} right)} right);quad left| {,left| x right|; to ;infty } right. cr
& Q(x,a) approx Q(x,1) - {{e^{, - 1} } over {Gamma (x)}}left( {a - 1} right) - {{e^{, - 1} left( {x - 2} right)} over {2Gamma (x)}}
left( {a - 1} right)^{,2} + Oleft( {left( {a - 1} right)^{,3} } right);quad left| {,left| a right|; to ;1} right. cr}
$$






share|cite|improve this answer























  • Thanks @G Cab. Is it possible to use Taylor here somehow?
    – Y.L
    Nov 28 at 23:34










  • do you mean a Taylor in $x$ ?
    – G Cab
    Nov 28 at 23:37










  • sorry there was a typo, corrected.
    – G Cab
    Nov 28 at 23:55










  • @G Cab, first of all big thank you! the lest term is beautiful. Intuitively though I was expecting the use of Taylor, and yes Taylor in $x$. Maybe there is a nice approximation for the last sum in your last a expression?
    – Y.L
    Nov 29 at 16:18










  • @Y.L: I added in the answer some considerations about this request. You should be more explicit about the values of $x$ and $a$ around which you want to develop in series.
    – G Cab
    Nov 29 at 22:51


















up vote
0
down vote













$$sum_{n=0}^infty ne^{-a}frac{a^{x+n}}{(x+n)!}=sum_{n=0}^infty frac{a^x}{e^a}frac{na^n}{(x+n)!}=frac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!};$$
$$scriptstylesum_{n=0}^infty a^nfrac{n}{(x+n)!}=e^aa^{1 - x} - frac{e^aa^{1 - x}Gamma(x + 1, a)}{Γ(x + 1)}- e^axa^{-x}+frac{e^aa^{-x}Gamma(x + 1, a)}{Gamma(x)} + frac{a}{Gamma(x + 1)}=e^aa^{1-x}-frac{e^aa^{1-x}}{x!}frac{x!}{e^a}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}-a^{1-x}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!};$$
$$scriptstylefrac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!}=frac{a^x}{e^a}left(e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!}right)$$
$$=a+left(frac{1}{x!}-frac{a}{e^a}right)sum_{k=0}^xfrac{a^k}{k!}-x+frac{a^{x+1}}{e^ax!}$$
That's the best I can do, it's a lot of algebra, so you might want to double check, but I think that's as simplified as it can possibley get.



This is not an algebraic function, so it can only really be written in terms of improper integrals or appropriate special functions.



https://www.wolframalpha.com/input/?i=%5Csum_%7Bn%3D0%7D%5E%5Cinfty+ne%5E%7B-a%7D%5Cfrac%7Ba%5E%7Bx%2Bn%7D%7D%7B(x%2Bn)!%7D






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    up vote
    0
    down vote













    Starting from R. Burton's answer, the result could simplify to
    $$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}=frac{a^{x+1} }{x!}left((x-a) E_{-x}(a)+e^{-a}right)+a-x$$ where appears the exponential integral function.






    share|cite|improve this answer





















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      3 Answers
      3






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      oldest

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      3 Answers
      3






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      oldest

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      active

      oldest

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      active

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      up vote
      2
      down vote



      accepted










      The sum can be written as
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} sumlimits_{k = x}^infty {left( {k - x} right){{a^{,k} } over {k!}}} = e^{, - ,a} left( {sumlimits_{k = 0}^infty
      {left( {k - x} right){{a^{,k} } over {k!}}} - sumlimits_{k = 0}^{x - 1} {left( {k - x} right){{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} - xsumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} + xsumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) cr}
      $$



      Since
      $$
      left{ matrix{
      e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = {{Gamma (x,a)} over {Gamma (x)}} = Q(x,a) hfill cr
      sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} = a{d over {da}}sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}} = ae^{,,a} hfill cr
      sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} = sumlimits_{k = 1}^{x - 1} {{{a^{,k} } over {left( {k - 1} right)!}}}
      = asumlimits_{j = 0}^{x - 2} {{{a^{,j} } over {j!}}} = ae^{,,a} Q(x - 1,a) hfill cr} right.
      $$



      where $Q(x,a)$ is the Regularized Incomplete Gamma function,

      it is easy to conclude.



      Another way is
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} left( {sumlimits_{n = 0}^infty {left( {n + x} right){{a^{,x + n} } over {left( {x + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,x + n - 1} } over {left( {x + n - 1} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n + 1} } over {left( {left( {x - 1} right) + n + 1} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + asumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} }
      over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{k = x}^infty {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + left( {a - x} right)sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - left( {a - x} right)sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} {{a^{,x} } over {left( {x - 1} right)!}} + left( {a - x} right)
      - left( {a - x} right)e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right)left( {1 - Q(x,a)} right) = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right){{gamma (x,a)} over {Gamma (x)}} cr}
      $$

      which checks with the previous one.



      --- Addendum ---



      Concerning your comment and the request for a nice approximation to the sum $S(a,x)$,
      it is not much clear what you intend / need to do. In any case consider the following:

      - the formula above (1st and 2nd way lead to the same result) is valid for $a$ and $c$
      real or even complex ( with some limitations);

      - if $x$ is an integer as you say, then for small values of it the expression in the last-but-third line
      is easily computable;

      - if $x$ is large , we can apply to $Gamma(x)$ and $Q(x,a)$ the known asymptotic expansions
      which are
      $$
      eqalign{
      & Gamma (x) = sqrt {,{{2,pi } over x},} left( {{x over e}} right)^{,x} left( {1 + Oleft( {{1 over x}} right)} right)
      quad left| {;x, to ,infty ,;;left| {,arg (x),} right|} right. < pi cr
      & Q(x,a) approx 1 - {{x^{, - x - 1/2} e^{,x - a} a^{,x} } over {sqrt {2pi } }}left( {1 + {{12a - 1} over {12x}}
      + Oleft( {{1 over {x^{,2} }}} right)} right);quad left| {,left| x right|; to ;infty } right. cr
      & Q(x,a) approx Q(x,1) - {{e^{, - 1} } over {Gamma (x)}}left( {a - 1} right) - {{e^{, - 1} left( {x - 2} right)} over {2Gamma (x)}}
      left( {a - 1} right)^{,2} + Oleft( {left( {a - 1} right)^{,3} } right);quad left| {,left| a right|; to ;1} right. cr}
      $$






      share|cite|improve this answer























      • Thanks @G Cab. Is it possible to use Taylor here somehow?
        – Y.L
        Nov 28 at 23:34










      • do you mean a Taylor in $x$ ?
        – G Cab
        Nov 28 at 23:37










      • sorry there was a typo, corrected.
        – G Cab
        Nov 28 at 23:55










      • @G Cab, first of all big thank you! the lest term is beautiful. Intuitively though I was expecting the use of Taylor, and yes Taylor in $x$. Maybe there is a nice approximation for the last sum in your last a expression?
        – Y.L
        Nov 29 at 16:18










      • @Y.L: I added in the answer some considerations about this request. You should be more explicit about the values of $x$ and $a$ around which you want to develop in series.
        – G Cab
        Nov 29 at 22:51















      up vote
      2
      down vote



      accepted










      The sum can be written as
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} sumlimits_{k = x}^infty {left( {k - x} right){{a^{,k} } over {k!}}} = e^{, - ,a} left( {sumlimits_{k = 0}^infty
      {left( {k - x} right){{a^{,k} } over {k!}}} - sumlimits_{k = 0}^{x - 1} {left( {k - x} right){{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} - xsumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} + xsumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) cr}
      $$



      Since
      $$
      left{ matrix{
      e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = {{Gamma (x,a)} over {Gamma (x)}} = Q(x,a) hfill cr
      sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} = a{d over {da}}sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}} = ae^{,,a} hfill cr
      sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} = sumlimits_{k = 1}^{x - 1} {{{a^{,k} } over {left( {k - 1} right)!}}}
      = asumlimits_{j = 0}^{x - 2} {{{a^{,j} } over {j!}}} = ae^{,,a} Q(x - 1,a) hfill cr} right.
      $$



      where $Q(x,a)$ is the Regularized Incomplete Gamma function,

      it is easy to conclude.



      Another way is
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} left( {sumlimits_{n = 0}^infty {left( {n + x} right){{a^{,x + n} } over {left( {x + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,x + n - 1} } over {left( {x + n - 1} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n + 1} } over {left( {left( {x - 1} right) + n + 1} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + asumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} }
      over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{k = x}^infty {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + left( {a - x} right)sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - left( {a - x} right)sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} {{a^{,x} } over {left( {x - 1} right)!}} + left( {a - x} right)
      - left( {a - x} right)e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right)left( {1 - Q(x,a)} right) = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right){{gamma (x,a)} over {Gamma (x)}} cr}
      $$

      which checks with the previous one.



      --- Addendum ---



      Concerning your comment and the request for a nice approximation to the sum $S(a,x)$,
      it is not much clear what you intend / need to do. In any case consider the following:

      - the formula above (1st and 2nd way lead to the same result) is valid for $a$ and $c$
      real or even complex ( with some limitations);

      - if $x$ is an integer as you say, then for small values of it the expression in the last-but-third line
      is easily computable;

      - if $x$ is large , we can apply to $Gamma(x)$ and $Q(x,a)$ the known asymptotic expansions
      which are
      $$
      eqalign{
      & Gamma (x) = sqrt {,{{2,pi } over x},} left( {{x over e}} right)^{,x} left( {1 + Oleft( {{1 over x}} right)} right)
      quad left| {;x, to ,infty ,;;left| {,arg (x),} right|} right. < pi cr
      & Q(x,a) approx 1 - {{x^{, - x - 1/2} e^{,x - a} a^{,x} } over {sqrt {2pi } }}left( {1 + {{12a - 1} over {12x}}
      + Oleft( {{1 over {x^{,2} }}} right)} right);quad left| {,left| x right|; to ;infty } right. cr
      & Q(x,a) approx Q(x,1) - {{e^{, - 1} } over {Gamma (x)}}left( {a - 1} right) - {{e^{, - 1} left( {x - 2} right)} over {2Gamma (x)}}
      left( {a - 1} right)^{,2} + Oleft( {left( {a - 1} right)^{,3} } right);quad left| {,left| a right|; to ;1} right. cr}
      $$






      share|cite|improve this answer























      • Thanks @G Cab. Is it possible to use Taylor here somehow?
        – Y.L
        Nov 28 at 23:34










      • do you mean a Taylor in $x$ ?
        – G Cab
        Nov 28 at 23:37










      • sorry there was a typo, corrected.
        – G Cab
        Nov 28 at 23:55










      • @G Cab, first of all big thank you! the lest term is beautiful. Intuitively though I was expecting the use of Taylor, and yes Taylor in $x$. Maybe there is a nice approximation for the last sum in your last a expression?
        – Y.L
        Nov 29 at 16:18










      • @Y.L: I added in the answer some considerations about this request. You should be more explicit about the values of $x$ and $a$ around which you want to develop in series.
        – G Cab
        Nov 29 at 22:51













      up vote
      2
      down vote



      accepted







      up vote
      2
      down vote



      accepted






      The sum can be written as
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} sumlimits_{k = x}^infty {left( {k - x} right){{a^{,k} } over {k!}}} = e^{, - ,a} left( {sumlimits_{k = 0}^infty
      {left( {k - x} right){{a^{,k} } over {k!}}} - sumlimits_{k = 0}^{x - 1} {left( {k - x} right){{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} - xsumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} + xsumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) cr}
      $$



      Since
      $$
      left{ matrix{
      e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = {{Gamma (x,a)} over {Gamma (x)}} = Q(x,a) hfill cr
      sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} = a{d over {da}}sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}} = ae^{,,a} hfill cr
      sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} = sumlimits_{k = 1}^{x - 1} {{{a^{,k} } over {left( {k - 1} right)!}}}
      = asumlimits_{j = 0}^{x - 2} {{{a^{,j} } over {j!}}} = ae^{,,a} Q(x - 1,a) hfill cr} right.
      $$



      where $Q(x,a)$ is the Regularized Incomplete Gamma function,

      it is easy to conclude.



      Another way is
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} left( {sumlimits_{n = 0}^infty {left( {n + x} right){{a^{,x + n} } over {left( {x + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,x + n - 1} } over {left( {x + n - 1} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n + 1} } over {left( {left( {x - 1} right) + n + 1} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + asumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} }
      over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{k = x}^infty {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + left( {a - x} right)sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - left( {a - x} right)sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} {{a^{,x} } over {left( {x - 1} right)!}} + left( {a - x} right)
      - left( {a - x} right)e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right)left( {1 - Q(x,a)} right) = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right){{gamma (x,a)} over {Gamma (x)}} cr}
      $$

      which checks with the previous one.



      --- Addendum ---



      Concerning your comment and the request for a nice approximation to the sum $S(a,x)$,
      it is not much clear what you intend / need to do. In any case consider the following:

      - the formula above (1st and 2nd way lead to the same result) is valid for $a$ and $c$
      real or even complex ( with some limitations);

      - if $x$ is an integer as you say, then for small values of it the expression in the last-but-third line
      is easily computable;

      - if $x$ is large , we can apply to $Gamma(x)$ and $Q(x,a)$ the known asymptotic expansions
      which are
      $$
      eqalign{
      & Gamma (x) = sqrt {,{{2,pi } over x},} left( {{x over e}} right)^{,x} left( {1 + Oleft( {{1 over x}} right)} right)
      quad left| {;x, to ,infty ,;;left| {,arg (x),} right|} right. < pi cr
      & Q(x,a) approx 1 - {{x^{, - x - 1/2} e^{,x - a} a^{,x} } over {sqrt {2pi } }}left( {1 + {{12a - 1} over {12x}}
      + Oleft( {{1 over {x^{,2} }}} right)} right);quad left| {,left| x right|; to ;infty } right. cr
      & Q(x,a) approx Q(x,1) - {{e^{, - 1} } over {Gamma (x)}}left( {a - 1} right) - {{e^{, - 1} left( {x - 2} right)} over {2Gamma (x)}}
      left( {a - 1} right)^{,2} + Oleft( {left( {a - 1} right)^{,3} } right);quad left| {,left| a right|; to ;1} right. cr}
      $$






      share|cite|improve this answer














      The sum can be written as
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} sumlimits_{k = x}^infty {left( {k - x} right){{a^{,k} } over {k!}}} = e^{, - ,a} left( {sumlimits_{k = 0}^infty
      {left( {k - x} right){{a^{,k} } over {k!}}} - sumlimits_{k = 0}^{x - 1} {left( {k - x} right){{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} - xsumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} + xsumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) cr}
      $$



      Since
      $$
      left{ matrix{
      e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = {{Gamma (x,a)} over {Gamma (x)}} = Q(x,a) hfill cr
      sumlimits_{k = 0}^infty {k{{a^{,k} } over {k!}}} = a{d over {da}}sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}} = ae^{,,a} hfill cr
      sumlimits_{k = 0}^{x - 1} {k{{a^{,k} } over {k!}}} = sumlimits_{k = 1}^{x - 1} {{{a^{,k} } over {left( {k - 1} right)!}}}
      = asumlimits_{j = 0}^{x - 2} {{{a^{,j} } over {j!}}} = ae^{,,a} Q(x - 1,a) hfill cr} right.
      $$



      where $Q(x,a)$ is the Regularized Incomplete Gamma function,

      it is easy to conclude.



      Another way is
      $$
      eqalign{
      & S(a,x) = sumlimits_{n = 0}^infty {ne^{, - ,a} {{a^{,x + n} } over {left( {x + n} right)!}}} = cr
      & = e^{, - ,a} left( {sumlimits_{n = 0}^infty {left( {n + x} right){{a^{,x + n} } over {left( {x + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,x + n - 1} } over {left( {x + n - 1} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n + 1} } over {left( {left( {x - 1} right) + n + 1} right)!}}} } right) = cr
      & = e^{, - ,a} left( {asumlimits_{n = 0}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + asumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} }
      over {left( {left( {x - 1} right) + n} right)!}}}
      - xsumlimits_{n = 1}^infty {{{a^{,left( {x - 1} right) + n} } over {left( {left( {x - 1} right) + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{n = 0}^infty {{{a^{,x + n} } over {left( {x + n} right)!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}}
      + left( {a - x} right)sumlimits_{k = x}^infty {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} left( {a{{a^{,left( {x - 1} right)} } over {left( {x - 1} right)!}} + left( {a - x} right)sumlimits_{k = 0}^infty {{{a^{,k} } over {k!}}}
      - left( {a - x} right)sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} } right) = cr
      & = e^{, - ,a} {{a^{,x} } over {left( {x - 1} right)!}} + left( {a - x} right)
      - left( {a - x} right)e^{, - ,a} sumlimits_{k = 0}^{x - 1} {{{a^{,k} } over {k!}}} = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right)left( {1 - Q(x,a)} right) = cr
      & = e^{, - ,a} {{a^{,x} } over {Gamma (x)}} + left( {a - x} right){{gamma (x,a)} over {Gamma (x)}} cr}
      $$

      which checks with the previous one.



      --- Addendum ---



      Concerning your comment and the request for a nice approximation to the sum $S(a,x)$,
      it is not much clear what you intend / need to do. In any case consider the following:

      - the formula above (1st and 2nd way lead to the same result) is valid for $a$ and $c$
      real or even complex ( with some limitations);

      - if $x$ is an integer as you say, then for small values of it the expression in the last-but-third line
      is easily computable;

      - if $x$ is large , we can apply to $Gamma(x)$ and $Q(x,a)$ the known asymptotic expansions
      which are
      $$
      eqalign{
      & Gamma (x) = sqrt {,{{2,pi } over x},} left( {{x over e}} right)^{,x} left( {1 + Oleft( {{1 over x}} right)} right)
      quad left| {;x, to ,infty ,;;left| {,arg (x),} right|} right. < pi cr
      & Q(x,a) approx 1 - {{x^{, - x - 1/2} e^{,x - a} a^{,x} } over {sqrt {2pi } }}left( {1 + {{12a - 1} over {12x}}
      + Oleft( {{1 over {x^{,2} }}} right)} right);quad left| {,left| x right|; to ;infty } right. cr
      & Q(x,a) approx Q(x,1) - {{e^{, - 1} } over {Gamma (x)}}left( {a - 1} right) - {{e^{, - 1} left( {x - 2} right)} over {2Gamma (x)}}
      left( {a - 1} right)^{,2} + Oleft( {left( {a - 1} right)^{,3} } right);quad left| {,left| a right|; to ;1} right. cr}
      $$







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Nov 29 at 22:47

























      answered Nov 28 at 22:39









      G Cab

      17.6k31237




      17.6k31237












      • Thanks @G Cab. Is it possible to use Taylor here somehow?
        – Y.L
        Nov 28 at 23:34










      • do you mean a Taylor in $x$ ?
        – G Cab
        Nov 28 at 23:37










      • sorry there was a typo, corrected.
        – G Cab
        Nov 28 at 23:55










      • @G Cab, first of all big thank you! the lest term is beautiful. Intuitively though I was expecting the use of Taylor, and yes Taylor in $x$. Maybe there is a nice approximation for the last sum in your last a expression?
        – Y.L
        Nov 29 at 16:18










      • @Y.L: I added in the answer some considerations about this request. You should be more explicit about the values of $x$ and $a$ around which you want to develop in series.
        – G Cab
        Nov 29 at 22:51


















      • Thanks @G Cab. Is it possible to use Taylor here somehow?
        – Y.L
        Nov 28 at 23:34










      • do you mean a Taylor in $x$ ?
        – G Cab
        Nov 28 at 23:37










      • sorry there was a typo, corrected.
        – G Cab
        Nov 28 at 23:55










      • @G Cab, first of all big thank you! the lest term is beautiful. Intuitively though I was expecting the use of Taylor, and yes Taylor in $x$. Maybe there is a nice approximation for the last sum in your last a expression?
        – Y.L
        Nov 29 at 16:18










      • @Y.L: I added in the answer some considerations about this request. You should be more explicit about the values of $x$ and $a$ around which you want to develop in series.
        – G Cab
        Nov 29 at 22:51
















      Thanks @G Cab. Is it possible to use Taylor here somehow?
      – Y.L
      Nov 28 at 23:34




      Thanks @G Cab. Is it possible to use Taylor here somehow?
      – Y.L
      Nov 28 at 23:34












      do you mean a Taylor in $x$ ?
      – G Cab
      Nov 28 at 23:37




      do you mean a Taylor in $x$ ?
      – G Cab
      Nov 28 at 23:37












      sorry there was a typo, corrected.
      – G Cab
      Nov 28 at 23:55




      sorry there was a typo, corrected.
      – G Cab
      Nov 28 at 23:55












      @G Cab, first of all big thank you! the lest term is beautiful. Intuitively though I was expecting the use of Taylor, and yes Taylor in $x$. Maybe there is a nice approximation for the last sum in your last a expression?
      – Y.L
      Nov 29 at 16:18




      @G Cab, first of all big thank you! the lest term is beautiful. Intuitively though I was expecting the use of Taylor, and yes Taylor in $x$. Maybe there is a nice approximation for the last sum in your last a expression?
      – Y.L
      Nov 29 at 16:18












      @Y.L: I added in the answer some considerations about this request. You should be more explicit about the values of $x$ and $a$ around which you want to develop in series.
      – G Cab
      Nov 29 at 22:51




      @Y.L: I added in the answer some considerations about this request. You should be more explicit about the values of $x$ and $a$ around which you want to develop in series.
      – G Cab
      Nov 29 at 22:51










      up vote
      0
      down vote













      $$sum_{n=0}^infty ne^{-a}frac{a^{x+n}}{(x+n)!}=sum_{n=0}^infty frac{a^x}{e^a}frac{na^n}{(x+n)!}=frac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!};$$
      $$scriptstylesum_{n=0}^infty a^nfrac{n}{(x+n)!}=e^aa^{1 - x} - frac{e^aa^{1 - x}Gamma(x + 1, a)}{Γ(x + 1)}- e^axa^{-x}+frac{e^aa^{-x}Gamma(x + 1, a)}{Gamma(x)} + frac{a}{Gamma(x + 1)}=e^aa^{1-x}-frac{e^aa^{1-x}}{x!}frac{x!}{e^a}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}-a^{1-x}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!};$$
      $$scriptstylefrac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!}=frac{a^x}{e^a}left(e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!}right)$$
      $$=a+left(frac{1}{x!}-frac{a}{e^a}right)sum_{k=0}^xfrac{a^k}{k!}-x+frac{a^{x+1}}{e^ax!}$$
      That's the best I can do, it's a lot of algebra, so you might want to double check, but I think that's as simplified as it can possibley get.



      This is not an algebraic function, so it can only really be written in terms of improper integrals or appropriate special functions.



      https://www.wolframalpha.com/input/?i=%5Csum_%7Bn%3D0%7D%5E%5Cinfty+ne%5E%7B-a%7D%5Cfrac%7Ba%5E%7Bx%2Bn%7D%7D%7B(x%2Bn)!%7D






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        $$sum_{n=0}^infty ne^{-a}frac{a^{x+n}}{(x+n)!}=sum_{n=0}^infty frac{a^x}{e^a}frac{na^n}{(x+n)!}=frac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!};$$
        $$scriptstylesum_{n=0}^infty a^nfrac{n}{(x+n)!}=e^aa^{1 - x} - frac{e^aa^{1 - x}Gamma(x + 1, a)}{Γ(x + 1)}- e^axa^{-x}+frac{e^aa^{-x}Gamma(x + 1, a)}{Gamma(x)} + frac{a}{Gamma(x + 1)}=e^aa^{1-x}-frac{e^aa^{1-x}}{x!}frac{x!}{e^a}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}-a^{1-x}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!};$$
        $$scriptstylefrac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!}=frac{a^x}{e^a}left(e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!}right)$$
        $$=a+left(frac{1}{x!}-frac{a}{e^a}right)sum_{k=0}^xfrac{a^k}{k!}-x+frac{a^{x+1}}{e^ax!}$$
        That's the best I can do, it's a lot of algebra, so you might want to double check, but I think that's as simplified as it can possibley get.



        This is not an algebraic function, so it can only really be written in terms of improper integrals or appropriate special functions.



        https://www.wolframalpha.com/input/?i=%5Csum_%7Bn%3D0%7D%5E%5Cinfty+ne%5E%7B-a%7D%5Cfrac%7Ba%5E%7Bx%2Bn%7D%7D%7B(x%2Bn)!%7D






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          up vote
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          $$sum_{n=0}^infty ne^{-a}frac{a^{x+n}}{(x+n)!}=sum_{n=0}^infty frac{a^x}{e^a}frac{na^n}{(x+n)!}=frac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!};$$
          $$scriptstylesum_{n=0}^infty a^nfrac{n}{(x+n)!}=e^aa^{1 - x} - frac{e^aa^{1 - x}Gamma(x + 1, a)}{Γ(x + 1)}- e^axa^{-x}+frac{e^aa^{-x}Gamma(x + 1, a)}{Gamma(x)} + frac{a}{Gamma(x + 1)}=e^aa^{1-x}-frac{e^aa^{1-x}}{x!}frac{x!}{e^a}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}-a^{1-x}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!};$$
          $$scriptstylefrac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!}=frac{a^x}{e^a}left(e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!}right)$$
          $$=a+left(frac{1}{x!}-frac{a}{e^a}right)sum_{k=0}^xfrac{a^k}{k!}-x+frac{a^{x+1}}{e^ax!}$$
          That's the best I can do, it's a lot of algebra, so you might want to double check, but I think that's as simplified as it can possibley get.



          This is not an algebraic function, so it can only really be written in terms of improper integrals or appropriate special functions.



          https://www.wolframalpha.com/input/?i=%5Csum_%7Bn%3D0%7D%5E%5Cinfty+ne%5E%7B-a%7D%5Cfrac%7Ba%5E%7Bx%2Bn%7D%7D%7B(x%2Bn)!%7D






          share|cite|improve this answer












          $$sum_{n=0}^infty ne^{-a}frac{a^{x+n}}{(x+n)!}=sum_{n=0}^infty frac{a^x}{e^a}frac{na^n}{(x+n)!}=frac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!};$$
          $$scriptstylesum_{n=0}^infty a^nfrac{n}{(x+n)!}=e^aa^{1 - x} - frac{e^aa^{1 - x}Gamma(x + 1, a)}{Γ(x + 1)}- e^axa^{-x}+frac{e^aa^{-x}Gamma(x + 1, a)}{Gamma(x)} + frac{a}{Gamma(x + 1)}=e^aa^{1-x}-frac{e^aa^{1-x}}{x!}frac{x!}{e^a}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}-a^{1-x}sum_{k=0}^{x}frac{a^k}{k!}-frac{e^ax}{a^x}+frac{e^a}{a^xx!}sum_{k=0}^{x}frac{a^k}{k!}+frac{a}{x!}=e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!};$$
          $$scriptstylefrac{a^x}{e^a}sum_{n=0}^infty a^nfrac{n}{(x+n)!}=frac{a^x}{e^a}left(e^aa^{1-x}+left(frac{e^a}{a^xx!}-a^{1-x}right)sum_{k=0}^xfrac{a^k}{k!}-frac{e^ax}{a^x}+frac{a}{x!}right)$$
          $$=a+left(frac{1}{x!}-frac{a}{e^a}right)sum_{k=0}^xfrac{a^k}{k!}-x+frac{a^{x+1}}{e^ax!}$$
          That's the best I can do, it's a lot of algebra, so you might want to double check, but I think that's as simplified as it can possibley get.



          This is not an algebraic function, so it can only really be written in terms of improper integrals or appropriate special functions.



          https://www.wolframalpha.com/input/?i=%5Csum_%7Bn%3D0%7D%5E%5Cinfty+ne%5E%7B-a%7D%5Cfrac%7Ba%5E%7Bx%2Bn%7D%7D%7B(x%2Bn)!%7D







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          share|cite|improve this answer










          answered Nov 28 at 23:50









          R. Burton

          3509




          3509






















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              Starting from R. Burton's answer, the result could simplify to
              $$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}=frac{a^{x+1} }{x!}left((x-a) E_{-x}(a)+e^{-a}right)+a-x$$ where appears the exponential integral function.






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                Starting from R. Burton's answer, the result could simplify to
                $$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}=frac{a^{x+1} }{x!}left((x-a) E_{-x}(a)+e^{-a}right)+a-x$$ where appears the exponential integral function.






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                  up vote
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                  Starting from R. Burton's answer, the result could simplify to
                  $$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}=frac{a^{x+1} }{x!}left((x-a) E_{-x}(a)+e^{-a}right)+a-x$$ where appears the exponential integral function.






                  share|cite|improve this answer












                  Starting from R. Burton's answer, the result could simplify to
                  $$sum_{n=0}^{infty}ne^{-a}frac{a^{x+n}}{(x+n)!}=frac{a^{x+1} }{x!}left((x-a) E_{-x}(a)+e^{-a}right)+a-x$$ where appears the exponential integral function.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 29 at 5:04









                  Claude Leibovici

                  118k1156131




                  118k1156131






























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