## Biorthogonal exponential sequences with weight function $\exp (ax^2+ibx)$ on the real line and an orthogonal sequence of trigonometric functions

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- by Mohammad Masjed-Jamei PDF
- Proc. Amer. Math. Soc.
**136**(2008), 409-417 Request permission

## Abstract:

Some orthogonal functions can be mapped onto other orthogonal functions by the Fourier transform. In this paper, by using the Fourier transform of StieltjesβWigert polynomials, we derive a sequence of exponential functions that are biorthogonal with respect to a complex weight function like $\exp (q_1(ix+p_1)^2+q_2(ix+p_2)^2)$ on $(-\infty ,\infty )$. Then we restrict these introduced biorthogonal functions to a special case to obtain a sequence of trigonometric functions orthogonal with respect to the real weight function $\exp (-qx^2)$ on $(-\infty ,\infty )$.## References

- T. S. Chihara,
*An introduction to orthogonal polynomials*, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR**0481884** - Jacob Stordal Christiansen,
*The moment problem associated with the Stieltjes-Wigert polynomials*, J. Math. Anal. Appl.**277**(2003), no.Β 1, 218β245. MR**1954473**, DOI 10.1016/S0022-247X(02)00534-6 - Jacob S. Christiansen and Erik Koelink,
*Self-adjoint difference operators and classical solutions to the Stieltjes-Wigert moment problem*, J. Approx. Theory**140**(2006), no.Β 1, 1β26. MR**2226673**, DOI 10.1016/j.jat.2005.11.010 - Jacob S. Christiansen and Mourad E. H. Ismail,
*A moment problem and a family of integral evaluations*, Trans. Amer. Math. Soc.**358**(2006), no.Β 9, 4071β4097. MR**2219011**, DOI 10.1090/S0002-9947-05-03785-2 - A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,
*Tables of Integral Transforms*, Vol. 2, McGraw-Hill, 1954. - Wolter Groenevelt,
*The Wilson function transform*, Int. Math. Res. Not.**52**(2003), 2779β2817. MR**2058035**, DOI 10.1155/S107379280313190X - M. E. H. Ismail and D. R. Masson,
*$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals*, Trans. Amer. Math. Soc.**346**(1994), no.Β 1, 63β116. MR**1264148**, DOI 10.1090/S0002-9947-1994-1264148-6 - R. Koekoek and R. F. Swarttouw,
*The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue*, Report no. 98-17, Technical Universiteit Delft, Faculty of Technical Mathematics and Informatics, Delft, (1998), Web site: http://aw.twi.tudelft.nl/ koekoek/askey/ - H. T. Koelink,
*On Jacobi and continuous Hahn polynomials*, Proc. Amer. Math. Soc.**124**(1996), no.Β 3, 887β898. MR**1307541**, DOI 10.1090/S0002-9939-96-03190-5 - W. Koepf and M. Masjed-Jamei,
*Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials*, Proc. Amer. Math. Soc.,**135**(2007), no. 11, 3599β3606. - T. H. Koornwinder,
*Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform*, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp.Β 174β183. MR**838982**, DOI 10.1007/BFb0076542 - Tom H. Koornwinder,
*Meixner-Pollaczek polynomials and the Heisenberg algebra*, J. Math. Phys.**30**(1989), no.Β 4, 767β769. MR**987105**, DOI 10.1063/1.528394 - Mohammad Masjedjamei,
*Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation*, Integral Transforms Spec. Funct.**13**(2002), no.Β 2, 169β191. MR**1915513**, DOI 10.1080/10652460212898 - Mohammad Masjed Jamei,
*Classical orthogonal polynomials with weight function $((ax+b)^2+(cx+d)^2)^{-p}\exp (q\,\textrm {Arctg}((ax+b)/(cx+d)))$, $x\in (-\infty ,\infty )$ and a generalization of $T$ and $F$ distributions*, Integral Transforms Spec. Funct.**15**(2004), no.Β 2, 137β153. MR**2053407**, DOI 10.1080/10652460410001663456 - T.J. Stieltjes,
*Recherches sur les fractions continues*, Annales de la faculte des sciences de Toulous,**8**(1894), J1β122;**9**(1895), A1β47; Qeuvres, vol.2, 398β566. - S. Wigert,
*Sur les polynomes orthogonaux et lβapproximation des functions continues*, Arkiv for matematik, astronomi och fysik,**17**(1923), no. 18, 15 pp.

## Additional Information

**Mohammad Masjed-Jamei**- Affiliation: Department of Applied Mathematics, K. N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran
- Email: mmjamei@aut.ac.ir, mmjamei@yahoo.com
- Received by editor(s): September 14, 2006
- Published electronically: November 1, 2007
- Communicated by: Carmen C. Chicone
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**136**(2008), 409-417 - MSC (2000): Primary 05E35, 42C05, 33C47
- DOI: https://doi.org/10.1090/S0002-9939-07-09139-3
- MathSciNet review: 2358478