Definition:qHahn: Difference between revisions

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:'''\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}''' produces <math>{\displaystyle \qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}}</math><br />
:'''\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}''' produces <math>{\displaystyle \qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}}</math><br />


These are defined by
These are defined by <ref>[[Formula:KLS:01.10:11]]</ref>
 
<math display=block>
\qHahn{N}@{q^{-x}}{\alpha}{\beta}{N}{q}=\sum_{k=0}^N
\frac{\qPochhammer{\alpha\beta q^{N+1}}{q}{k}\qPochhammer{q^{-x}}{q}{k}}{\qPochhammer{\alpha q}{q}{k}\qPochhammer{q}{q}{k}}q^k
</math>
 
or <ref>[[Formula:KLS:14.06:01]]</ref>
 
<math>{\displaystyle
<math>{\displaystyle
\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}:=\qHyperrphis{3}{2}@@{q^{-n},\alpha\beta q^{n+1},q^{-x}}{\alpha q,q^{-N}}{q}{q}
\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}:=\qHyperrphis{3}{2}@@{q^{-n},\alpha\beta q^{n+1},q^{-x}}{\alpha q,q^{-N}}{q}{q}

Revision as of 04:04, 14 July 2017

The LaTeX DLMF and DRMF macro \qHahn represents the q 𝑞 {\displaystyle{\displaystyle q}} -Hahn polynomial.

This macro is in the category of polynomials.

In math mode, this macro can be called in the following ways:

\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N} produces Q n ⁡ ( q - x ; α , β , N ; ) q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 absent {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};\alpha,\beta,N;% \right)}}}
\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N} produces ⁢ Q n ⁡ ( q - x ) q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 absent q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle}{Q_{n}\!\left(q^{-x}\right)}}}

These are defined by [1]

Q N ⁡ ( q - x ; α , β , N ; q ) = ∑ k = 0 N ( α ⁢ β ⁢ q N + 1 ; q ) k ⁢ ( q - x ; q ) k ( α ⁢ q ; q ) k ⁢ ( q ; q ) k ⁢ q k q-Hahn-polynomial-Q 𝑁 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 superscript subscript 𝑘 0 𝑁 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 𝑁 1 𝑞 𝑘 q-Pochhammer-symbol superscript 𝑞 𝑥 𝑞 𝑘 q-Pochhammer-symbol 𝛼 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 superscript 𝑞 𝑘 {\displaystyle Q_{N}\!\left(q^{-x};\alpha,\beta,N;q\right)=\sum_{k=0}^{N}\frac% {\left(\alpha\beta q^{N+1};q\right)_{k}\left(q^{-x};q\right)_{k}}{\left(\alpha q% ;q\right)_{k}\left(q;q\right)_{k}}q^{k}}

or [2]

Q n ⁡ ( q - x ; α , β , N ; q ) := \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , α ⁢ β ⁢ q n + 1 , q - x ⁢ α ⁢ q , q - N ⁢ q ⁢ q assign q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 superscript 𝑞 𝑥 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};\alpha,\beta,N;% q\right):=\qHyperrphis{3}{2}@@{q^{-n},\alpha\beta q^{n+1},q^{-x}}{\alpha q,q^{% -N}}{q}{q}}}}

Symbols List

Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:qHahn
ϕ s r subscript subscript italic-ϕ 𝑠 𝑟 {\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}  : basic hypergeometric (or q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -hypergeometric) function : http://dlmf.nist.gov/17.4#E1


  1. ↑ Formula:KLS:01.10:11
  2. ↑ Formula:KLS:14.06:01
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