How to show that $frac{4^n}{n^{3/2}sqrt pi}$ could not be expressed as $sum_i^m p_i(n)lambda_i^n$
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How to show that
$$
frac{4^n}{n^{3/2}sqrt pi}
$$ has not the form $p_1(n) lambda_1^n + ldots p_i(n) lambda_i^{n}$ for some polynomials $p_i(n)$ and numbers $lambda_i$?
real-analysis combinatorics complex-analysis number-theory analysis
asked Nov 21 at 16:45
StefanH
8,04552161
add a comment |
up vote
0
down vote
favorite
How to show that
$$
frac{4^n}{n^{3/2}sqrt pi}
$$ has not the form $p_1(n) lambda_1^n + ldots p_i(n) lambda_i^{n}$ for some polynomials $p_i(n)$ and numbers $lambda_i$?
real-analysis combinatorics complex-analysis number-theory analysis
asked Nov 21 at 16:45
StefanH
8,04552161
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How to show that
$$
frac{4^n}{n^{3/2}sqrt pi}
$$ has not the form $p_1(n) lambda_1^n + ldots p_i(n) lambda_i^{n}$ for some polynomials $p_i(n)$ and numbers $lambda_i$?
real-analysis combinatorics complex-analysis number-theory analysis
asked Nov 21 at 16:45
StefanH
8,04552161
How to show that
$$
frac{4^n}{n^{3/2}sqrt pi}
$$ has not the form $p_1(n) lambda_1^n + ldots p_i(n) lambda_i^{n}$ for some polynomials $p_i(n)$ and numbers $lambda_i$?
real-analysis combinatorics complex-analysis number-theory analysis
real-analysis combinatorics complex-analysis number-theory analysis
asked Nov 21 at 16:45
StefanH
8,04552161
asked Nov 21 at 16:45
StefanH
8,04552161
asked Nov 21 at 16:45
StefanH
8,04552161
asked Nov 21 at 16:45
StefanH
8,04552161
asked Nov 21 at 16:45
StefanH
8,04552161
8,04552161
add a comment |
add a comment |
1 Answer
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up vote
3
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The Laplace transform of $p(x) K^x$ is a rational function whose singularities are poles.
The Laplace transform of $frac{4^x}{x^{3/2}}$ has a branch point (of the $Csqrt{s-2log 2}$ kind) at $s=2log 2$, hence $frac{4^x}{x^{3/2}}$ and $sum p_m(x) K_m^x$ cannot be the same function.
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
Just because this was mentioned without further explanation in a basic textbook. Is there a more elementary way to see it?
– StefanH
Nov 21 at 18:04
Jack: do you think you can show that not only the sequence is not rational, but that it's power series is not $D$-finite, in the kernel of some differential operator with polynomial coefficients?
– Pedro Tamaroff♦
Nov 21 at 18:13
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
The Laplace transform of $p(x) K^x$ is a rational function whose singularities are poles.
The Laplace transform of $frac{4^x}{x^{3/2}}$ has a branch point (of the $Csqrt{s-2log 2}$ kind) at $s=2log 2$, hence $frac{4^x}{x^{3/2}}$ and $sum p_m(x) K_m^x$ cannot be the same function.
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
Just because this was mentioned without further explanation in a basic textbook. Is there a more elementary way to see it?
– StefanH
Nov 21 at 18:04
Jack: do you think you can show that not only the sequence is not rational, but that it's power series is not $D$-finite, in the kernel of some differential operator with polynomial coefficients?
– Pedro Tamaroff♦
Nov 21 at 18:13
add a comment |
up vote
3
down vote
The Laplace transform of $p(x) K^x$ is a rational function whose singularities are poles.
The Laplace transform of $frac{4^x}{x^{3/2}}$ has a branch point (of the $Csqrt{s-2log 2}$ kind) at $s=2log 2$, hence $frac{4^x}{x^{3/2}}$ and $sum p_m(x) K_m^x$ cannot be the same function.
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
Just because this was mentioned without further explanation in a basic textbook. Is there a more elementary way to see it?
– StefanH
Nov 21 at 18:04
Jack: do you think you can show that not only the sequence is not rational, but that it's power series is not $D$-finite, in the kernel of some differential operator with polynomial coefficients?
– Pedro Tamaroff♦
Nov 21 at 18:13
add a comment |
up vote
3
down vote
up vote
3
down vote
The Laplace transform of $p(x) K^x$ is a rational function whose singularities are poles.
The Laplace transform of $frac{4^x}{x^{3/2}}$ has a branch point (of the $Csqrt{s-2log 2}$ kind) at $s=2log 2$, hence $frac{4^x}{x^{3/2}}$ and $sum p_m(x) K_m^x$ cannot be the same function.
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
The Laplace transform of $p(x) K^x$ is a rational function whose singularities are poles.
The Laplace transform of $frac{4^x}{x^{3/2}}$ has a branch point (of the $Csqrt{s-2log 2}$ kind) at $s=2log 2$, hence $frac{4^x}{x^{3/2}}$ and $sum p_m(x) K_m^x$ cannot be the same function.
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
answered Nov 21 at 16:51
Jack D'Aurizio
284k33275654
284k33275654
Just because this was mentioned without further explanation in a basic textbook. Is there a more elementary way to see it?
– StefanH
Nov 21 at 18:04
Jack: do you think you can show that not only the sequence is not rational, but that it's power series is not $D$-finite, in the kernel of some differential operator with polynomial coefficients?
– Pedro Tamaroff♦
Nov 21 at 18:13
add a comment |
Just because this was mentioned without further explanation in a basic textbook. Is there a more elementary way to see it?
– StefanH
Nov 21 at 18:04
Jack: do you think you can show that not only the sequence is not rational, but that it's power series is not $D$-finite, in the kernel of some differential operator with polynomial coefficients?
– Pedro Tamaroff♦
Nov 21 at 18:13
Just because this was mentioned without further explanation in a basic textbook. Is there a more elementary way to see it?
– StefanH
Nov 21 at 18:04
Just because this was mentioned without further explanation in a basic textbook. Is there a more elementary way to see it?
– StefanH
Nov 21 at 18:04
Jack: do you think you can show that not only the sequence is not rational, but that it's power series is not $D$-finite, in the kernel of some differential operator with polynomial coefficients?
– Pedro Tamaroff♦
Nov 21 at 18:13
Jack: do you think you can show that not only the sequence is not rational, but that it's power series is not $D$-finite, in the kernel of some differential operator with polynomial coefficients?
– Pedro Tamaroff♦
Nov 21 at 18:13
add a comment |
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