Formula:KLS:09.03:04

From DRMF
Jump to navigation Jump to search


- ( a 2 + x 2 ) ⁢ S ~ n ⁡ ( x 2 ) = A n ⁢ S ~ n + 1 ⁡ ( x 2 ) - ( A n + C n ) ⁢ S ~ n ⁡ ( x 2 ) + C n ⁢ S ~ n - 1 ⁡ ( x 2 ) superscript 𝑎 2 superscript 𝑥 2 continuous-dual-Hahn-normalized-S-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 subscript 𝐴 𝑛 continuous-dual-Hahn-normalized-S-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-dual-Hahn-normalized-S-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 subscript 𝐶 𝑛 continuous-dual-Hahn-normalized-S-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle-\left(a^{2}+x^{2}\right){\tilde{S}}% _{n}\!\left(x^{2}\right)=A_{n}{\tilde{S}}_{n+1}\!\left(x^{2}\right)-\left(A_{n% }+C_{n}\right){\tilde{S}}_{n}\!\left(x^{2}\right)+C_{n}{\tilde{S}}_{n-1}\!% \left(x^{2}\right)}}}

Substitution(s)

C n = n ⁢ ( n + b + c - 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝑏 𝑐 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=n(n+b+c-1)}}} &
A n = ( n + a + b ) ⁢ ( n + a + c ) subscript 𝐴 𝑛 𝑛 𝑎 𝑏 𝑛 𝑎 𝑐 {\displaystyle{\displaystyle{\displaystyle A_{n}=(n+a+b)(n+a+c)}}}


Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
S ~ n subscript ~ 𝑆 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{S}}_{n}}}}  : normalized continuous dual Hahn polynomial S ~ ~ 𝑆 {\displaystyle{\displaystyle{\displaystyle{\tilde{S}}}}}  : http://drmf.wmflabs.org/wiki/Definition:normctsdualHahnStilde

Bibliography

Equation in Section 9.3 of KLS.

URL links

We ask users to provide relevant URL links in this space.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:09.03:04&oldid=3397"