Connections between Laguerre polynomials through a third-order differential operator transformation
Volume 39, Issue 2, pp 292--299
https://dx.doi.org/10.22436/jmcs.039.02.08
Publication Date: March 29, 2025
Submission Date: January 25, 2025
Revision Date: February 10, 2025
Accteptance Date: February 25, 2025
Authors
W. Chammam
- Department of Mathematics, College of Science At Zulfi, Majmaah University, Al Majmaah, 11952, Saudi Arabia.
B. Aloui
- Department of Mathematics, Faculty of Sciences Sfax, Sfax University, Sfax, Tunisia.
- Research Laboratory Mathematics and Applications LR17ES11, Gabes University, Erriadh City, 6072 Zrig, Gabes, Tunisia.
- University of Carthage, National Institute of Applied Sciences and Technology, Tunisia.
J. Souissi
- Department of Mathematics, Faculty of Sciences Gabes, Gabes University, Gabes, Tunisia.
Abstract
Let \(\{l^{(\alpha)}_n\}_{n\geq 0}\), (\(\alpha\neq-m, m\geq1\)),
be the monic orthogonal sequence of Laguerre polynomials.
We define a new differential operator,
\(\mathscr{L}^{+}_{\alpha}\), that raises the degree and also the parameter of
\(l^{(\alpha)}_n(x)\). More precisely,
\(\mathscr{L}^{+}_{\alpha}l^{(\alpha)}_n(x)=l^{(\alpha+1)}_{n+1}(x), n\geq0\).
As an illustration, we give some properties related to
this operator and some other operators in the literature.
Share and Cite
ISRP Style
W. Chammam, B. Aloui, J. Souissi, Connections between Laguerre polynomials through a third-order differential operator transformation, Journal of Mathematics and Computer Science, 39 (2025), no. 2, 292--299
AMA Style
Chammam W., Aloui B., Souissi J., Connections between Laguerre polynomials through a third-order differential operator transformation. J Math Comput SCI-JM. (2025); 39(2):292--299
Chicago/Turabian Style
Chammam, W., Aloui, B., Souissi, J.. "Connections between Laguerre polynomials through a third-order differential operator transformation." Journal of Mathematics and Computer Science, 39, no. 2 (2025): 292--299
Keywords
- Laguerre polynomials
- differential operators
- lowering-raising-shift operators
- eigenfunctions
MSC
References
-
[1]
M. Alfaro, J. S. Dehesa, F. J. Marcellán, J. L. Rubio, J. Vinuesa, Orthogonal Polynomials and their Applications, In: Lecture Notes in Mathematics, Springer-Verlag, Berlin (1988)
-
[2]
B. Aloui, Characterization of Laguerre polynomials as orthogonal polynomials connected by the Laguerre degree raising shift operator, Ramanujan J., 45 (2018), 475–481
-
[3]
B. Aloui, Hermite polynomials and Hahn’s theorem with respect to the raising operator, Electron. J. Math. Anal. Appl., 6 (2018), 157–162
-
[4]
B. Aloui, Chebyshev polynomials of the second kind via raising operator preserving the orthogonality, Period. Math. Hungar., 76 (2018), 126–132
-
[5]
B. Aloui, W. Chammam, Classical orthogonal polynomials and some new properties for their centroids of zeroes, Anal. Math., 46 (202), 13–23
-
[6]
B. Aloui, W. Chammam, Classical orthogonal polynomials via a second-order linear differential operators, Indian J. Pure Appl. Math., 51 (2020), 689–703
-
[7]
B. Aloui, W. Chammam, Z. A. Alhussain, The centroid of the zeroes of a polynomial via certain Laguerre-type operators, Math. Methods Appl. Sci., 43 (2020), 2656–2664
-
[8]
B. Aloui, L. Khériji, Connection formulas and representations of Laguerre polynomials in terms of the action of linear differential operators, Probl. Anal. Issues Anal., 8(26) (2019), 24–37
-
[9]
W. A. Al-Salam, Characterization theorems for orthogonal polynomials, Kluwer Acad. Publ., Dordrecht (1990)
-
[10]
W. Al-Salam, T. S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal., 3 (1972), 65–70
-
[11]
A. I. Aptekarev, J. S. Dehesa, R. J. Yáñez, Spatial entropy of central potentials and strong asymptotics of orthogonal polynomials, J. Math. Phys., 35 (1994), 4423–4428
-
[12]
S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z., 29 (1929), 730–736
-
[13]
H. Chaggara, Operational rules and a generalized Hermite polynomials, J. Math. Anal. Appl., 332 (2007), 11–21
-
[14]
W. Chammam, W. Ul-Haq, Some new connection relations related to classical orthogonal polynomials, J. Math., 2020 (2020), 6 pages
-
[15]
T. Chen, F. Li, P. Yu, Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysis, J. Differ. Equ., 379 (2024), 258–289
-
[16]
T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London- Paris (1978)
-
[17]
J. S. Dehesa, Orthogonal polynomials in neutron transport theories, J. Phys. A: Math. Gen., 15 (1982), 327–330
-
[18]
J. S. Dehesa, F. Domínguez-Adame, E. R. Arriola, A. Zarzo, Hydrogen atom and orthogonal polynomials, In: Orthogonal polynomials and their applications (Erice, 1990), Baltzer, Basel, 9 (1991), 223–229
-
[19]
J. S. Dehesa, M. Van Assche, R. J. Yáñez, Information entropy of classical orthogonal polynomials and their application to the harmonic oscillator and Coulomb potentials, Methods Appl. Anal., 4 (1997), 91–110
-
[20]
T. H. Koornwinder, Lowering and raising operators for some special orthogonal polynomials, Amer. Math. Soc., Providence, RI, 417 (2006), 227–238
-
[21]
N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York (1972)
-
[22]
A. F. Loureiro, P. Maroni, Quadratic decomposition of Appell sequences, Expo. Math., 26 (2008), 177–186
-
[23]
P. Maroni, Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, In: Orthogonal polynomials and their applications (Erice, 1990), Baltzer, Basel, 9 (1991), 95–130
-
[24]
P. Maroni, Fonctions Eulériennes, Polynômes Orthogonaux Classiques, Téch. l’Ingén., Traité Gén. (Sci. Fondam.), A 154 (1994), 1–30
-
[25]
J. Petronilho, On the linear functionals associated to linearly related sequences of orthogonal polynomials, J. Math. Anal. Appl., 315 (2006), 379–393
-
[26]
E. D. Rainville, Special functions, The Macmillan Company, New York (1960)
-
[27]
P. E. Ricci, Laguerre-Type Exponentials, Laguerre Derivatives and Applications. A Survey, Mathematics, 8 (2020), 17 pages
-
[28]
J. Souissi, Characterization of polynomials via a raising operator, Probl. Anal. Issues Anal., 13(31) (2024), 71–81
-
[29]
H. M. Srivastava, Y. Ben Cheikh, Orthogonality of some polynomial sets via quasi-monomiality, Appl. Math. Comput., 141 (2003), 415–425
-
[30]
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, Rhode Island (1975)
-
[31]
Y. Xue, J. Han, Z. Tu, X. Chen, Stability analysis and design of cooperative control for linear delta operator system, AIMS Math., 8 (2023), 12671–12693
-
[32]
A. Zarzo, J. S. Dehesa, A. Ronveaux, Newton sum rules of zeros of semiclassical orthogonal polynomials, J. Comput. Appl. Math., 33 (1990), 85–96
