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q-difference polynomial
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<h2 id="definition">Definition</h2> <p>The q-difference polynomials satisfy the relation </p> ( d d z ) q p n ( z ) = p n ( q z ) − p n ( z ) q z − z = q n − 1 q − 1 p n − 1 ( z ) = [ n ] q p n − 1 ( z ) {\displaystyle \left({\frac {d}{dz}}\right)_{q}p_{n}(z)={\frac {p_{n}(qz)-p_{n}(z)}{qz-z}}={\frac {q^{n}-1}{q-1}}p_{n-1}(z)=[n]_{q}p_{n-1}(z)} <p>where the derivative symbol on the left is the q-derivative. In the limit of q → 1 {\displaystyle q\to 1} , this becomes the definition of the Appell polynomials: </p> d d z p n ( z ) = n p n − 1 ( z ) . {\displaystyle {\frac {d}{dz}}p_{n}(z)=np_{n-1}(z).} <h2 id="generating-function">Generating function</h2> <p>The generalized <a href="/facts/Generating_function/K7UVjnjL">generating function</a> for these polynomials is of the type of generating function for Brenke polynomials, namely </p> A ( w ) e q ( z w ) = ∑ n = 0 ∞ p n ( z ) [ n ] q ! w n {\displaystyle A(w)e_{q}(zw)=\sum _{n=0}^{\infty }{\frac {p_{n}(z)}{[n]_{q}!}}w^{n}} <p>where e q ( t ) {\displaystyle e_{q}(t)} is the <a href="/facts/Q-exponential/t9gJxytF">q-exponential</a>: </p> e q ( t ) = ∑ n = 0 ∞ t n [ n ] q ! = ∑ n = 0 ∞ t n ( 1 − q ) n ( q ; q ) n . {\displaystyle e_{q}(t)=\sum _{n=0}^{\infty }{\frac {t^{n}}{[n]_{q}!}}=\sum _{n=0}^{\infty }{\frac {t^{n}(1-q)^{n}}{(q;q)_{n}}}.} <p>Here, [ n ] q ! {\displaystyle [n]_{q}!} is the <a href="/facts/Q-factorial/keASLfpe">q-factorial</a> and </p> ( q ; q ) n = ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q ) {\displaystyle (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)} <p>is the <a href="/facts/Q-Pochhammer_symbol/keASLfpe">q-Pochhammer symbol</a>. The function A ( w ) {\displaystyle A(w)} is arbitrary but assumed to have an expansion </p> A ( w ) = ∑ n = 0 ∞ a n w n with a 0 ≠ 0. {\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}{\mbox{ with }}a_{0}\neq 0.} <p>Any such A ( w ) {\displaystyle A(w)} gives a sequence of q-difference polynomials. </p> <ul><li>A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", <i>Riv. Mat. Univ. Parma</i>, 5 (1954) 325–337.</li> <li>Ralph P. Boas, Jr. and R. Creighton Buck, <i>Polynomial Expansions of Analytic Functions (Second Printing Corrected)</i>, (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. <i>(Provides a very brief discussion of convergence.)</i></li></ul>