DLMF:17.2.E42 (Q5332): Difference between revisions

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Revision as of 14:51, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 14:51, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint
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		Property / Symbols used
			Q11298
		Property / Symbols used: Q11298 / rank
			Normal rank
		Property / Symbols used: Q11298 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$ \qPochhammer{\NVar{a}}{\NVar{q}}{\NVar{n}}
		Property / Symbols used: Q11298 / qualifier
			xml-id: C17.S2.SS1.p1.m2aaidec
		Property / Symbols used
			$$q$$ -binomial coefficient (or Gaussian polynomial)
		Property / Symbols used: $$q$$ -binomial coefficient (or Gaussian polynomial) / rank
			Normal rank
		Property / Symbols used: $$q$$ -binomial coefficient (or Gaussian polynomial) / qualifier
			Defining formula: ${\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}}}$ \qbinom{\NVar{n}}{\NVar{m}}{\NVar{q}}
		Property / Symbols used: $$q$$ -binomial coefficient (or Gaussian polynomial) / qualifier
			xml-id: C17.S2.E27.m2aldec
		Property / Symbols used
			Q11710
		Property / Symbols used: Q11710 / rank
			Normal rank
		Property / Symbols used: Q11710 / qualifier
			Defining formula: ${\displaystyle{\displaystyle q}}$ q
		Property / Symbols used: Q11710 / qualifier
			xml-id: C17.S1.XMD10.m1andec
		Property / Symbols used
			Q11712
		Property / Symbols used: Q11712 / rank
			Normal rank
		Property / Symbols used: Q11712 / qualifier
			Defining formula: ${\displaystyle{\displaystyle j}}$ j
		Property / Symbols used: Q11712 / qualifier
			xml-id: C17.S1.XMD2.m1idec
		Property / Symbols used
			Q11711
		Property / Symbols used: Q11711 / rank
			Normal rank
		Property / Symbols used: Q11711 / qualifier
			Defining formula: ${\displaystyle{\displaystyle n}}$ n
		Property / Symbols used: Q11711 / qualifier
			xml-id: C17.S1.XMD4.m1akdec
		Property / Symbols used
			Q11718
		Property / Symbols used: Q11718 / rank
			Normal rank
		Property / Symbols used: Q11718 / qualifier
			Defining formula: ${\displaystyle{\displaystyle z}}$ z
		Property / Symbols used: Q11718 / qualifier
			xml-id: C17.S1.XMD7.m1edec
		Property / Symbols used
			$$q$$ -differential operator (locally)
		Property / Symbols used: $$q$$ -differential operator (locally) / rank
			Normal rank
		Property / Symbols used: $$q$$ -differential operator (locally) / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathcal{D}_{q}}}$ \mathcal{D}_{q}
		Property / Symbols used: $$q$$ -differential operator (locally) / qualifier
			xml-id: C17.S2.XMD1.m1adec

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Latest revision as of 14:51, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:17.2.E42	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle f^{[n]}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{-n}(1-q)^{-n}\sum_{j=0}^{n}q^{-nj+\genfrac{(}{)}{0.0pt}{}{j+1}{2}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}f(zq^{j}),&z\neq 0,\\ \dfrac{f^{(n)}(0)\left(q;q\right)_{n}}{n!(1-q)^{n}},&z=0.\end{cases}}}$

0 references

DLMF id

DLMF:17.2.E42

0 references

Symbols used

Q10754

Defining formula

${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}$

xml-id

C1.S2.SS1.m1amdec

0 references

Q10755

Defining formula

${\displaystyle{\displaystyle!}}$

xml-id

introduction.Sx4.p1.t1.r15.m5aadec

0 references

Q11298

Defining formula

${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$

xml-id

C17.S2.SS1.p1.m2aaidec

0 references

$$q$$ -binomial coefficient (or Gaussian polynomial)

Defining formula

${\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}}}$

xml-id

C17.S2.E27.m2aldec

0 references

Q11710

Defining formula

${\displaystyle{\displaystyle q}}$

xml-id

C17.S1.XMD10.m1andec

0 references

Q11712

Defining formula

${\displaystyle{\displaystyle j}}$

xml-id

C17.S1.XMD2.m1idec

0 references

Q11711

Defining formula

${\displaystyle{\displaystyle n}}$

xml-id

C17.S1.XMD4.m1akdec

0 references

Q11718

Defining formula

${\displaystyle{\displaystyle z}}$

xml-id

C17.S1.XMD7.m1edec

0 references

$$q$$ -differential operator (locally)

Defining formula

${\displaystyle{\displaystyle\mathcal{D}_{q}}}$

xml-id

C17.S2.XMD1.m1adec

0 references

 

Sitelinks

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