Formula:KLS:14.22:05

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- x ⁢ y n ⁡ ( x ; a ; q ) = A n ⁢ y n + 1 ⁡ ( x ; a ; q ) - ( A n + C n ) ⁢ y n ⁡ ( x ; a ; q ) + C n ⁢ y n - 1 ⁡ ( x ; a ; q ) 𝑥 q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 q-Bessel-polynomial-y 𝑛 1 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 subscript 𝐶 𝑛 q-Bessel-polynomial-y 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle-xy_{n}\!\left(x;a;q\right)=A_{n}y_{% n+1}\!\left(x;a;q\right)-(A_{n}+C_{n})y_{n}\!\left(x;a;q\right)+C_{n}y_{n-1}\!% \left(x;a;q\right)}}}

Substitution(s)

C n = a ⁢ q 2 ⁢ n - 1 ⁢ ( 1 - q n ) ( 1 + a ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 + a ⁢ q 2 ⁢ n ) subscript 𝐶 𝑛 𝑎 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 1 𝑎 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{2n-1}\frac{(1-q^{n})}{(1+% aq^{2n-1})(1+aq^{2n})}}}} &
A n = q n ⁢ ( 1 + a ⁢ q n ) ( 1 + a ⁢ q 2 ⁢ n ) ⁢ ( 1 + a ⁢ q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1+aq^{n})}{(1+aq^% {2n})(1+aq^{2n+1})}}}}


Proof

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Symbols List

& : logical and
y n subscript 𝑦 𝑛 {\displaystyle{\displaystyle{\displaystyle y_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Bessel polynomial : http://drmf.wmflabs.org/wiki/Definition:qBessel

Bibliography

Equation in Section 14.22 of KLS.

URL links

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