Q-derivative power rule
From specialfunctionswiki
Theorem
The following formula holds: $$D_q(z^n)=[n]_q z^{n-1},$$ where $D_q$ denotes the $q$-derivative and $[n]_q$ denotes the $q$-number.
Proof
References
- {{ #if: |{{{2}}}|D.S. McAnally}}{{#if: |{{#if: |, [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]}}|}}: [[Paper:D.S. McAnally/q-exponential and q-gamma functions. I. q-exponential functions{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|q-exponential and q-gamma functions. I. q-exponential functions{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}} (1994)| ({{#if: |{{{ed}}} ed., }}1994)}}]]{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: q-Derivative | ... (previous)|}}{{#if: q-number | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}} $(2.2)$
