Formula:KLS:09.08:43

From DRMF
Jump to navigation Jump to search


R n ( α , α ) ⁡ ( x ⁢ y + ( 1 - x 2 ) 1 2 ⁢ ( 1 - y 2 ) 1 2 ⁢ t ) = ∑ k = 0 n ( - 1 ) k ⁢ ( - n ) k ⁢ ( n + 2 ⁢ α + 1 ) k 2 2 ⁢ k ⁢ ( ( α + 1 ) k ) 2 ⁢ ( 1 - x 2 ) k / 2 ⁢ R n - k ( α + k , α + k ) ⁡ ( x ) ⁢ ( 1 - y 2 ) k / 2 ⁢ R n - k ( α + k , α + k ) ⁡ ( y ) ⁢ ω k ( α - 1 2 , α - 1 2 ) ⁢ R k ( α - 1 2 , α - 1 2 ) ⁡ ( t ) normalized-Jacobi-polynomial-R 𝛼 𝛼 𝑛 𝑥 𝑦 superscript 1 superscript 𝑥 2 1 2 superscript 1 superscript 𝑦 2 1 2 𝑡 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 Pochhammer-symbol 𝑛 𝑘 Pochhammer-symbol 𝑛 2 𝛼 1 𝑘 superscript 2 2 𝑘 superscript Pochhammer-symbol 𝛼 1 𝑘 2 superscript 1 superscript 𝑥 2 𝑘 2 normalized-Jacobi-polynomial-R 𝛼 𝑘 𝛼 𝑘 𝑛 𝑘 𝑥 superscript 1 superscript 𝑦 2 𝑘 2 normalized-Jacobi-polynomial-R 𝛼 𝑘 𝛼 𝑘 𝑛 𝑘 𝑦 superscript subscript 𝜔 𝑘 𝛼 1 2 𝛼 1 2 normalized-Jacobi-polynomial-R 𝛼 1 2 𝛼 1 2 𝑘 𝑡 {\displaystyle{\displaystyle{\displaystyle R^{(\alpha,\alpha)}_{n}\left(xy+(1-% x^{2})^{\frac{1}{2}}(1-y^{2})^{\frac{1}{2}}t\right)=\sum_{k=0}^{n}\frac{(-1)^{% k}{\left(-n\right)_{k}}{\left(n+2\alpha+1\right)_{k}}}{2^{2k}({\left(\alpha+1% \right)_{k}})^{2}}(1-x^{2})^{k/2}R^{(\alpha+k,\alpha+k)}_{n-k}\left(x\right)(1% -y^{2})^{k/2}R^{(\alpha+k,\alpha+k)}_{n-k}\left(y\right)\omega_{k}^{(\alpha-% \frac{1}{2},\alpha-\frac{1}{2})}R^{(\alpha-\frac{1}{2},\alpha-\frac{1}{2})}_{k% }\left(t\right)}}}

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

R n ( α , β ) subscript superscript 𝑅 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle R^{(\alpha,\beta)}_{n}}}}  : normalized Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:normJacobiR
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.8 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:09.08:43&oldid=3715"