Formula:KLS:14.22:08

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x ⁢ y ^ n ⁡ ( x ) = y ^ n + 1 ⁡ ( x ) + ( A n + C n ) ⁢ y ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ y ^ n - 1 ⁡ ( x ) 𝑥 q-Bessel-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 q-Bessel-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Bessel-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Bessel-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{y}}_{n}\!\left(x\right)=% {\widehat{y}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}){\widehat{y}}_{n}\!\left(x% \right)+A_{n-1}C_{n}{\widehat{y}}_{n-1}\!\left(x\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{\widehat{y}}_{n}}}}  : monic q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Bessel polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqBessel

Bibliography

Equation in Section 14.22 of KLS.

URL links

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