The \(L\)-Appell symmetric orthogonal polynomials. (English) Zbl 1376.33012
Summary: In this paper, we shall be concerned with lowering operators defined on polynomials by means of
\[
L(x^n)=\mu_nx^{n-1},\quad n=0,1,\dots,\quad \mu_0=0,\quad \mu_n\neq 0\quad (n=1,2,\dots).
\]
We determine a necessary and sufficient condition on lowering operators \(L\) and a symmetric orthogonal polynomial sets \(\{P_n\}_{n\geq 0}\) such that \(\{P_n\}_{n\geq 0}\) is \(L\)-Appell. The resulting polynomials are the generalized Hermite and the symmetric PSs related to Wall and generalized Stieltjes-Wigert. Various properties of the obtained families are singled out: a three-term recurrence relation, explicit expression in term of hypergeometric and basic hypergeometric functions and generating functions.
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
orthogonal polynomials; symmetric orthogonal polynomials; \(L\)-Appell polynomials; lowering operators; generalized Hermite polynomialsReferences:
| [1] | Angelesco, A.: Sur les polynômes orthogonaux en rapport avec d’autres polynômes. Bul. Soc. Stiite Cluj. 1, 44-59 (1921) · JFM 48.0423.02 |
| [2] | Al-Salam, W.A.: \[q\] q-Appell polynomials. Ann. Mat. Pura Appl. 77, 31-45 (1967) · Zbl 0157.38901 · doi:10.1007/BF02416939 |
| [3] | Appell, P.: Sur une classe de polynômes. Ann. Sci. École Norm. Sup. 9, 119-144 (1880) · JFM 12.0342.02 · doi:10.24033/asens.186 |
| [4] | Ben Cheikh, Y., Gaied, M.: Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl. Math. Comput. 187, 105-114 (2007) · Zbl 1112.33004 |
| [5] | Ben Cheikh, Y., Gaied, M.: Dunkl-Appell \[d\] d-orthogonal polynomials. Integral Transforms Spec. Func. 18, 581-597 (2007) · Zbl 1137.42005 · doi:10.1080/10652460701445302 |
| [6] | Ben Cheikh, Y., Gaied, M.: \[q\] q-Dunkl-classical q-Hermite type polynomials. Georgian Math. J. 21, 125-137 (2014) · Zbl 1296.33017 · doi:10.1515/gmj-2014-0022 |
| [7] | Bouanani, A., Khériji, L., Ihsen Tounsi, M.: Characterization of q-Dunkl Appell symmetric orthogonal q-polynomials. Expo. Math. 28, 325-336 (2010) · Zbl 1204.33008 · doi:10.1016/j.exmath.2010.03.003 |
| [8] | Chihara, T.S.: Orthogonal polynomials with Brenke type generating functions. Duke Math. J. 35, 505-518 (1968) · Zbl 0174.10902 · doi:10.1215/S0012-7094-68-03551-5 |
| [9] | Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordan and Breach, New York (1978) · Zbl 0389.33008 |
| [10] | Ghressi, A., Khériji, L.: On the q-analogue of Dunkl operator and its Appell classical orthogonal polynomials. Int. J. Pure Appl. Math. Sci. 39, 1-16 (2007) · Zbl 1148.42006 |
| [11] | Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and their q-analogues Springer Monographs in Mathematics. Springer, Berlin (2010) · Zbl 1200.33012 · doi:10.1007/978-3-642-05014-5 |
| [12] | Meixner, J.: Orthogonal polynomsysteme mit einer besondern gestalt der erzeugender functionen. J. Lond. Math. Soc. 9, 6-13 (1934) · JFM 60.0293.01 · doi:10.1112/jlms/s1-9.1.6 |
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