Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Generalized Appell polynomials
open-in-new
<h2 id="special-cases">Special cases</h2> <ul><li>The choice of g ( w ) = w {\displaystyle g(w)=w} gives the class of <a href="/facts/Brenke_polynomials/VLeKNJ4G">Brenke polynomials</a>.</li> <li>The choice of Ψ ( t ) = e t {\displaystyle \Psi (t)=e^{t}} results in the <a href="/facts/Sheffer_sequence/CpjBBWcM">Sheffer sequence</a> of polynomials, which include the <a href="/facts/General_difference_polynomials/Dd09q2NC">general difference polynomials</a>, such as the <a href="/facts/Newton_polynomials/2vRsjpA2">Newton polynomials</a>.</li> <li>The combined choice of g ( w ) = w {\displaystyle g(w)=w} and Ψ ( t ) = e t {\displaystyle \Psi (t)=e^{t}} gives the <a href="/facts/Appell_sequence/7hEeFjzM">Appell sequence</a> of polynomials.</li></ul> <h2 id="explicit-representation">Explicit representation</h2> <p>The generalized Appell polynomials have the explicit representation </p> p n ( z ) = ∑ k = 0 n z k Ψ k h k . {\displaystyle p_{n}(z)=\sum _{k=0}^{n}z^{k}\Psi _{k}h_{k}.} <p>The constant is </p> h k = ∑ P a j 0 g j 1 g j 2 ⋯ g j k {\displaystyle h_{k}=\sum _{P}a_{j_{0}}g_{j_{1}}g_{j_{2}}\cdots g_{j_{k}}} <p>where this sum extends over all <a href="/facts/Composition_(combinatorics)/sfBgx7Cd">compositions</a> of n {\displaystyle n} into k + 1 {\displaystyle k+1} parts; that is, the sum extends over all { j } {\displaystyle \{j\}} such that </p> j 0 + j 1 + ⋯ + j k = n . {\displaystyle j_{0}+j_{1}+\cdots +j_{k}=n.\,} <p>For the Appell polynomials, this becomes the formula </p> p n ( z ) = ∑ k = 0 n a n − k z k k ! . {\displaystyle p_{n}(z)=\sum _{k=0}^{n}{\frac {a_{n-k}z^{k}}{k!}}.} <h2 id="recursion-relation">Recursion relation</h2> <p>Equivalently, a necessary and sufficient condition that the kernel K ( z , w ) {\displaystyle K(z,w)} can be written as A ( w ) Ψ ( z g ( w ) ) {\displaystyle A(w)\Psi (zg(w))} with g 1 = 1 {\displaystyle g_{1}=1} is that </p> ∂ K ( z , w ) ∂ w = c ( w ) K ( z , w ) + z b ( w ) w ∂ K ( z , w ) ∂ z {\displaystyle {\frac {\partial K(z,w)}{\partial w}}=c(w)K(z,w)+{\frac {zb(w)}{w}}{\frac {\partial K(z,w)}{\partial z}}} <p>where b ( w ) {\displaystyle b(w)} and c ( w ) {\displaystyle c(w)} have the <a href="/facts/Power_series/vYh6sTps">power series</a> </p> b ( w ) = w g ( w ) d d w g ( w ) = 1 + ∑ n = 1 ∞ b n w n {\displaystyle b(w)={\frac {w}{g(w)}}{\frac {d}{dw}}g(w)=1+\sum _{n=1}^{\infty }b_{n}w^{n}} <p>and </p> c ( w ) = 1 A ( w ) d d w A ( w ) = ∑ n = 0 ∞ c n w n . {\displaystyle c(w)={\frac {1}{A(w)}}{\frac {d}{dw}}A(w)=\sum _{n=0}^{\infty }c_{n}w^{n}.} <p>Substituting </p> K ( z , w ) = ∑ n = 0 ∞ p n ( z ) w n {\displaystyle K(z,w)=\sum _{n=0}^{\infty }p_{n}(z)w^{n}} <p>immediately gives the <a href="/facts/Recursion_relation/JolXC71M">recursion relation</a> </p> z n + 1 d d z [ p n ( z ) z n ] = − ∑ k = 0 n − 1 c n − k − 1 p k ( z ) − z ∑ k = 1 n − 1 b n − k d d z p k ( z ) . {\displaystyle z^{n+1}{\frac {d}{dz}}\left[{\frac {p_{n}(z)}{z^{n}}}\right]=-\sum _{k=0}^{n-1}c_{n-k-1}p_{k}(z)-z\sum _{k=1}^{n-1}b_{n-k}{\frac {d}{dz}}p_{k}(z).} <p>For the special case of the Brenke polynomials, one has g ( w ) = w {\displaystyle g(w)=w} and thus all of the b n = 0 {\displaystyle b_{n}=0} , simplifying the recursion relation significantly. </p> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/Q-difference_polynomial/EXfI08XU">q-difference polynomials</a></li></ul> <ul><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.</li> <li>Brenke, William C. (1945). "On generating functions of polynomial systems". <i>American Mathematical Monthly</i>. 52 (6): 297–301. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2305289">10.2307/2305289</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2305289">2305289</a>.</li> <li>Huff, W. N. (1947). "The type of the polynomials generated by f(xt) φ(t)". <i>Duke Mathematical Journal</i>. 14 (4): 1091–1104. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2FS0012-7094-47-01483-X">10.1215/S0012-7094-47-01483-X</a>.</li></ul>