Do you recognize this infinite series? $sum_{n=0}^infty frac 1{(1+an)^c} frac{b^n}{n!} $
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2
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I came by this infinite series
$$sum_{n=0}^infty frac 1{(1+an)^c} frac{b^n}{n!} $$
Is there some special function that can have this form?
$c$ can be assumed to be a positive integer. While $a,b$ are positive real numbers
calculus power-series special-functions
asked Nov 27 at 23:59
M.A
1529
add a comment |
up vote
2
down vote
favorite
I came by this infinite series
$$sum_{n=0}^infty frac 1{(1+an)^c} frac{b^n}{n!} $$
Is there some special function that can have this form?
$c$ can be assumed to be a positive integer. While $a,b$ are positive real numbers
calculus power-series special-functions
asked Nov 27 at 23:59
M.A
1529
Thanks this is interesting. But I am interested in the case where a is not necessarily integer.
– M.A
Nov 28 at 0:09
Related: en.wikipedia.org/wiki/Hurwitz_zeta_function
– Henricus V.
Nov 28 at 0:13
1
The sum is equal to $$ frac{1}{a^c Gamma(c)} int_{0}^{1} e^{bx} x^{(1/a)-1} log^{c-1}(1/x) , dx. $$ I am not sure if this simplifies further...
– Sangchul Lee
Nov 28 at 0:13
1
If there is any $f(x)$ that satisfies $f^{(n)}(0)=(1+an)^{-c}$, then the sum is equal to $f(b)$
– Seth
Nov 28 at 0:17
1
When $c$ is a positive integer, the expression equals to a generalized hypergeometric function ${}_cF_cleft( alpha_1,ldots,alpha_c ; beta_1,ldots,beta_c; bright)$ where all $alpha_i = frac1a$ and and all $beta_i = frac1a + 1$.
– achille hui
Nov 28 at 0:26
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I came by this infinite series
$$sum_{n=0}^infty frac 1{(1+an)^c} frac{b^n}{n!} $$
Is there some special function that can have this form?
$c$ can be assumed to be a positive integer. While $a,b$ are positive real numbers
calculus power-series special-functions
asked Nov 27 at 23:59
M.A
1529
I came by this infinite series
$$sum_{n=0}^infty frac 1{(1+an)^c} frac{b^n}{n!} $$
Is there some special function that can have this form?
$c$ can be assumed to be a positive integer. While $a,b$ are positive real numbers
calculus power-series special-functions
calculus power-series special-functions
asked Nov 27 at 23:59
M.A
1529
asked Nov 27 at 23:59
M.A
1529
edited Nov 28 at 0:11
asked Nov 27 at 23:59
M.A
1529
asked Nov 27 at 23:59
M.A
1529
asked Nov 27 at 23:59
M.A
1529
1529
Thanks this is interesting. But I am interested in the case where a is not necessarily integer.
– M.A
Nov 28 at 0:09
Related: en.wikipedia.org/wiki/Hurwitz_zeta_function
– Henricus V.
Nov 28 at 0:13
1
The sum is equal to $$ frac{1}{a^c Gamma(c)} int_{0}^{1} e^{bx} x^{(1/a)-1} log^{c-1}(1/x) , dx. $$ I am not sure if this simplifies further...
– Sangchul Lee
Nov 28 at 0:13
1
If there is any $f(x)$ that satisfies $f^{(n)}(0)=(1+an)^{-c}$, then the sum is equal to $f(b)$
– Seth
Nov 28 at 0:17
1
When $c$ is a positive integer, the expression equals to a generalized hypergeometric function ${}_cF_cleft( alpha_1,ldots,alpha_c ; beta_1,ldots,beta_c; bright)$ where all $alpha_i = frac1a$ and and all $beta_i = frac1a + 1$.
– achille hui
Nov 28 at 0:26
add a comment |
Thanks this is interesting. But I am interested in the case where a is not necessarily integer.
– M.A
Nov 28 at 0:09
Related: en.wikipedia.org/wiki/Hurwitz_zeta_function
– Henricus V.
Nov 28 at 0:13
1
The sum is equal to $$ frac{1}{a^c Gamma(c)} int_{0}^{1} e^{bx} x^{(1/a)-1} log^{c-1}(1/x) , dx. $$ I am not sure if this simplifies further...
– Sangchul Lee
Nov 28 at 0:13
1
If there is any $f(x)$ that satisfies $f^{(n)}(0)=(1+an)^{-c}$, then the sum is equal to $f(b)$
– Seth
Nov 28 at 0:17
1
When $c$ is a positive integer, the expression equals to a generalized hypergeometric function ${}_cF_cleft( alpha_1,ldots,alpha_c ; beta_1,ldots,beta_c; bright)$ where all $alpha_i = frac1a$ and and all $beta_i = frac1a + 1$.
– achille hui
Nov 28 at 0:26
Thanks this is interesting. But I am interested in the case where a is not necessarily integer.
– M.A
Nov 28 at 0:09
Thanks this is interesting. But I am interested in the case where a is not necessarily integer.
– M.A
Nov 28 at 0:09
Related: en.wikipedia.org/wiki/Hurwitz_zeta_function
– Henricus V.
Nov 28 at 0:13
Related: en.wikipedia.org/wiki/Hurwitz_zeta_function
– Henricus V.
Nov 28 at 0:13
1
1
The sum is equal to $$ frac{1}{a^c Gamma(c)} int_{0}^{1} e^{bx} x^{(1/a)-1} log^{c-1}(1/x) , dx. $$ I am not sure if this simplifies further...
– Sangchul Lee
Nov 28 at 0:13
The sum is equal to $$ frac{1}{a^c Gamma(c)} int_{0}^{1} e^{bx} x^{(1/a)-1} log^{c-1}(1/x) , dx. $$ I am not sure if this simplifies further...
– Sangchul Lee
Nov 28 at 0:13
1
1
If there is any $f(x)$ that satisfies $f^{(n)}(0)=(1+an)^{-c}$, then the sum is equal to $f(b)$
– Seth
Nov 28 at 0:17
If there is any $f(x)$ that satisfies $f^{(n)}(0)=(1+an)^{-c}$, then the sum is equal to $f(b)$
– Seth
Nov 28 at 0:17
1
1
When $c$ is a positive integer, the expression equals to a generalized hypergeometric function ${}_cF_cleft( alpha_1,ldots,alpha_c ; beta_1,ldots,beta_c; bright)$ where all $alpha_i = frac1a$ and and all $beta_i = frac1a + 1$.
– achille hui
Nov 28 at 0:26
When $c$ is a positive integer, the expression equals to a generalized hypergeometric function ${}_cF_cleft( alpha_1,ldots,alpha_c ; beta_1,ldots,beta_c; bright)$ where all $alpha_i = frac1a$ and and all $beta_i = frac1a + 1$.
– achille hui
Nov 28 at 0:26
add a comment |
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Thanks this is interesting. But I am interested in the case where a is not necessarily integer.
– M.A
Nov 28 at 0:09
Related: en.wikipedia.org/wiki/Hurwitz_zeta_function
– Henricus V.
Nov 28 at 0:13
1
The sum is equal to $$ frac{1}{a^c Gamma(c)} int_{0}^{1} e^{bx} x^{(1/a)-1} log^{c-1}(1/x) , dx. $$ I am not sure if this simplifies further...
– Sangchul Lee
Nov 28 at 0:13
1
If there is any $f(x)$ that satisfies $f^{(n)}(0)=(1+an)^{-c}$, then the sum is equal to $f(b)$
– Seth
Nov 28 at 0:17
1
When $c$ is a positive integer, the expression equals to a generalized hypergeometric function ${}_cF_cleft( alpha_1,ldots,alpha_c ; beta_1,ldots,beta_c; bright)$ where all $alpha_i = frac1a$ and and all $beta_i = frac1a + 1$.
– achille hui
Nov 28 at 0:26