Formula:KLS:09.03:16

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( a - 1 2 - i ⁢ x ) ⁢ ( b - 1 2 - i ⁢ x ) ⁢ ( c - 1 2 - i ⁢ x ) ⁢ S n ⁡ ( ( x + 1 2 ⁢ i ) 2 ; a , b , c ) - ( a - 1 2 + i ⁢ x ) ⁢ ( b - 1 2 + i ⁢ x ) ⁢ ( c - 1 2 + i ⁢ x ) ⁢ S n ⁡ ( ( x - 1 2 ⁢ i ) 2 ; a , b , c ) = - 2 ⁢ i ⁢ x ⁢ S n + 1 ⁢ ( x 2 ; a - 1 2 , b - 1 2 ⁢ c - 1 2 ) 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 2 imaginary-unit 𝑥 subscript 𝑆 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 {\displaystyle{\displaystyle{\displaystyle(a-\textstyle\frac{1}{2}-\mathrm{i}x% )(b-\textstyle\frac{1}{2}-\mathrm{i}x)(c-\textstyle\frac{1}{2}-\mathrm{i}x)S_{% n}\!\left((x+\textstyle\frac{1}{2}\mathrm{i})^{2};a,b,c\right){}-(a-\textstyle% \frac{1}{2}+\mathrm{i}x)(b-\textstyle\frac{1}{2}+\mathrm{i}x)(c-\textstyle% \frac{1}{2}+\mathrm{i}x)S_{n}\!\left((x-\textstyle\frac{1}{2}\mathrm{i})^{2};a% ,b,c\right){}=-2\mathrm{i}xS_{n+1}(x^{2};a-\textstyle\frac{1}{2},b-\textstyle% \frac{1}{2}c-\textstyle\frac{1}{2})}}}

Proof

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Symbols List

i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
S n subscript 𝑆 𝑛 {\displaystyle{\displaystyle{\displaystyle S_{n}}}}  : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3

Bibliography

Equation in Section 9.3 of KLS.

URL links

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