Fast Hypergeometric representation of Error function
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I been checking the following representation
$$sum _{k=0}^n frac{2^{2 k+2} sqrt{p^2} (-1)^{k-n} Gamma left(k+frac{5}{2}right) Gamma (k+n+2) , _3F_3left(frac{1}{2},frac{k}{2}+frac{5}{4},frac{k}{2}+frac{7}{4};frac{3}{2},frac{k}{2}+frac{3}{2},frac{k}{2}+2;-p^2right)}{pi Gamma (k+3) Gamma (2 k+2) Gamma (-k+n+1)}=text{Erf}(p)$$ as n tends to infinity is faster than Taylor and Burmann series , it is possible to show it ?? it seems to be the same series representation as Taylor series
calculus
edited Jan 8 at 23:38
Bernard
119k639112
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
$endgroup$
add a comment |
$begingroup$
I been checking the following representation
$$sum _{k=0}^n frac{2^{2 k+2} sqrt{p^2} (-1)^{k-n} Gamma left(k+frac{5}{2}right) Gamma (k+n+2) , _3F_3left(frac{1}{2},frac{k}{2}+frac{5}{4},frac{k}{2}+frac{7}{4};frac{3}{2},frac{k}{2}+frac{3}{2},frac{k}{2}+2;-p^2right)}{pi Gamma (k+3) Gamma (2 k+2) Gamma (-k+n+1)}=text{Erf}(p)$$ as n tends to infinity is faster than Taylor and Burmann series , it is possible to show it ?? it seems to be the same series representation as Taylor series
calculus
edited Jan 8 at 23:38
Bernard
119k639112
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
$endgroup$
$begingroup$
Where did you find it ?
$endgroup$
– Claude Leibovici
Jan 9 at 5:29
$begingroup$
the Special Functions and Their Approximations: Volume 1 [Yudell L. Luke]
$endgroup$
– CLERKRAMA
Jan 9 at 9:29
add a comment |
$begingroup$
I been checking the following representation
$$sum _{k=0}^n frac{2^{2 k+2} sqrt{p^2} (-1)^{k-n} Gamma left(k+frac{5}{2}right) Gamma (k+n+2) , _3F_3left(frac{1}{2},frac{k}{2}+frac{5}{4},frac{k}{2}+frac{7}{4};frac{3}{2},frac{k}{2}+frac{3}{2},frac{k}{2}+2;-p^2right)}{pi Gamma (k+3) Gamma (2 k+2) Gamma (-k+n+1)}=text{Erf}(p)$$ as n tends to infinity is faster than Taylor and Burmann series , it is possible to show it ?? it seems to be the same series representation as Taylor series
calculus
edited Jan 8 at 23:38
Bernard
119k639112
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
$endgroup$
I been checking the following representation
$$sum _{k=0}^n frac{2^{2 k+2} sqrt{p^2} (-1)^{k-n} Gamma left(k+frac{5}{2}right) Gamma (k+n+2) , _3F_3left(frac{1}{2},frac{k}{2}+frac{5}{4},frac{k}{2}+frac{7}{4};frac{3}{2},frac{k}{2}+frac{3}{2},frac{k}{2}+2;-p^2right)}{pi Gamma (k+3) Gamma (2 k+2) Gamma (-k+n+1)}=text{Erf}(p)$$ as n tends to infinity is faster than Taylor and Burmann series , it is possible to show it ?? it seems to be the same series representation as Taylor series
calculus
calculus
edited Jan 8 at 23:38
Bernard
119k639112
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
edited Jan 8 at 23:38
Bernard
119k639112
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
edited Jan 8 at 23:38
Bernard
119k639112
edited Jan 8 at 23:38
Bernard
119k639112
edited Jan 8 at 23:38
Bernard
119k639112
119k639112
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
asked Jan 8 at 23:23
CLERKRAMACLERKRAMA
84
84
$begingroup$
Where did you find it ?
$endgroup$
– Claude Leibovici
Jan 9 at 5:29
$begingroup$
the Special Functions and Their Approximations: Volume 1 [Yudell L. Luke]
$endgroup$
– CLERKRAMA
Jan 9 at 9:29
add a comment |
$begingroup$
Where did you find it ?
$endgroup$
– Claude Leibovici
Jan 9 at 5:29
$begingroup$
the Special Functions and Their Approximations: Volume 1 [Yudell L. Luke]
$endgroup$
– CLERKRAMA
Jan 9 at 9:29
$begingroup$
Where did you find it ?
$endgroup$
– Claude Leibovici
Jan 9 at 5:29
$begingroup$
Where did you find it ?
$endgroup$
– Claude Leibovici
Jan 9 at 5:29
$begingroup$
the Special Functions and Their Approximations: Volume 1 [Yudell L. Luke]
$endgroup$
– CLERKRAMA
Jan 9 at 9:29
$begingroup$
the Special Functions and Their Approximations: Volume 1 [Yudell L. Luke]
$endgroup$
– CLERKRAMA
Jan 9 at 9:29
add a comment |
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$begingroup$
Where did you find it ?
$endgroup$
– Claude Leibovici
Jan 9 at 5:29
$begingroup$
the Special Functions and Their Approximations: Volume 1 [Yudell L. Luke]
$endgroup$
– CLERKRAMA
Jan 9 at 9:29