Geodesic
Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type
Communications in Mathematics, Tome 28 (2020) no. 1, pp. 13-25 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.
In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.
Classification : 34B40, 34L10, 39A12, 39A13, 39A70
Keywords: Hahn's Sturm-Liouville equation; spectral function; Parseval equality; spectral expansion. 
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Allahverdiev, Bilender P.; Tuna, Hüseyin. Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type. Communications in Mathematics, Tome 28 (2020) no. 1, pp. 13-25. http://geodesic.mathdoc.fr/item/COMIM_2020_28_1_a1/

[1] Aldwoah, K.A.: Generalized time scales and associated difference equations. 2009, Cairo University, Ph.D. Thesis.

[2] Allahverdiev, B.P., Tuna, H.: An expansion theorem for $q$-Sturm-Liouville operators on the whole line. Turkish J. Math., 42, 3, 2018, 1060-1071, | MR

[3] Allahverdiev, B.P., Tuna, H.: Spectral expansion for singular Dirac system with impulsive conditions. Turkish J. Math., 42, 5, 2018, 2527-2545, | DOI | MR

[4] Allahverdiev, B.P., Tuna, H.: Eigenfunction expansion in the singular case for Dirac systems on time scales. Konuralp J. Math., 7, 1, 2019, 128-135, | MR

[5] Allahverdiev, B.P., Tuna, H.: The spectral expansion for Hahn-Dirac system on the whole line. Turkish J. Math., 43, 2019, 1668-1687, | DOI | MR

[6] Allahverdiev, B.P., Tuna, H.: Eigenfunction expansion for singular Sturm-Liouville problems with transmission conditions. Electron. J. Differ.Equat., 2019, 3, 2019, 1-10, | MR

[7] Allahverdiev, B.P., Tuna, H.: The Parseval equality and expansion formula for singular Hahn-Dirac system. Emerging Applications of Differential Equations and Game Theory, 2020, 209-235, IGI Global,

[8] Álvarez-Nodarse, R.: On characterizations of classical polynomials. J. Comput. Appl. Math., 196, 1, 2006, 320-337, | DOI | MR

[9] Annaby, M.H., Hamza, A.E., Aldwoah, K.A.: Hahn difference operator and associated Jackson-Nörlund integrals. J. Optim. Theory Appl., 154, 2012, 133-153, | DOI | MR

[10] Annaby, M.H., Hamza, A.E., Makharesh, S.D.: A Sturm-Liouville theory for Hahn difference operator. Frontiers of Orthogonal Polynomials and $q$-Series, 2018, 35-84, World Scientific, Singapore, | MR

[11] Annaby, M.A., Hassan, H.A.: Sampling theorems forJackson-Nörlund transforms associated with Hahn-difference operators. J. Math. Anal. Appl., 464, 1, 2018, 493-506, | DOI | MR

[12] Arvesú, J.: On some properties of $q-$Hahn multiple orthogonal polynomials. J. Comput. Appl. Math., 233, 6, 2010, 1462-1469, Elsevier, doi:10.1016/j.cam.2009.02.062. | DOI | MR

[13] Berezanskii, J.M.: Expansions in Eigenfunctions of Selfadjoint Operators. 1968, Amer. Math. Soc., Providence, | MR | Zbl

[14] Dobrogowska, A., Odzijewicz, A.: Second order $q$-difference equations solvable by factorization method. J. Comput. Appl. Math., 193, 1, 2006, 319-346, | DOI | MR

[15] Guseinov, G.Sh.: Eigenfunction expansions for a Sturm-Liouville problem on time scales. Int. J. Difference Equat., 2, 1, 2007, 93-104, | MR

[16] Guseinov, G.Sh.: An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl., 3, 1, 2008, 147-160, | MR

[17] Hahn, W.: Über orthogonalpolynome, die $q$-Differenzengleichungen genügen. Math. Nachr., 2, 1949, 4-34, | DOI | MR

[18] Hahn, W.: Ein Beitrag zur Theorie der Orthogonalpolynome. Monatsh. Math., 95, 1983, 19-24, | DOI | MR

[19] Hamza, A.E., Ahmed, S.A.: Existence and uniqueness of solutions of Hahn difference equations. Adv. Difference Equat., 316, 2013, 1-15, | MR

[20] Hamza, A.E., Makharesh, S.D.: Leibniz' rule and Fubinis theorem associated with Hahn difference operator. J. Adv. Math., 12, 6, 2016, 6335-6345, | DOI

[21] Huseynov, A., Bairamov, E.: On expansions in eigenfunctions for second order dynamic equations on time scales. Nonlinear Dyn. Syst. Theory, 9, 1, 2009, 77-88, | MR

[22] Huseynov, A.: Eigenfunction expansion associated with the one-dimensional Schrödinger equation on semi-infinite time scale intervals. Rep. Math. Phys., 66, 2, 2010, 207-235, | DOI | MR

[23] Jackson, F.H.: $q$-Difference equations. Amer. J. Math., 32, 1910, 305-314, | DOI | MR

[24] Jagerman, D.L.: Difference Equations with Applications to Queues. 2000, Dekker, New York, | MR

[25] Jordan, C.: Calculus of Finite Differences, 3rd edn. 1965, Chelsea, New York, | MR

[26] Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Translated by R.A. Silverman. 1970, Dover Publications, New York, | MR

[27] Kwon, K.H., Lee, D.W., Park, S.B., Yoo, B.H.: Hahn class orthogonal polynomials. Kyungpook Math. J., 38, 1998, 259-281, | MR

[28] Lesky, P.A.: Eine Charakterisierung der klassischen kontinuierlichen, diskretenund $q$-Orthgonalpolynome. 2005, Shaker, Aachen,

[29] Levinson, N.: A simplified proof of the expansion theorem for singular second order linear differential equations. Duke Math. J., 18, 1951, 57-71, | DOI | MR

[30] Levitan, B.M., Sargsjan, I.S.: Sturm-Liouville and Dirac Operators. 1991, Springer, | MR

[31] Naimark, M.A.: Linear Differential Operators, 2nd edn., 1968. 1969, Nauka, Moscow, English translation of 1st edn.. | MR

[32] Petronilho, J.: Generic formulas for the values at the singular points of some special monic classical $H_{q,\omega }$-orthogonal polynomials. J. Comput. Appl. Math., 205, 2007, 314-324, | DOI | MR

[33] Sitthiwirattham, T.: On a nonlocal boundary value problem for nonlinear second-order Hahn difference equation with two different $q,\omega $-derivatives. Adv. Difference Equat., 2016, 1, 2016, Article number 116. | MR

[34] Stone, M.H.: A comparison of the series of Fourier and Birkhoff. Trans. Amer. Math. Soc., 28, 1926, 695-761, | DOI | MR

[35] Stone, M.H.: Linear Transformations in Hilbert Space and Their Application to Analysis. 1932, Amer. Math. Soc., | MR

[36] Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition. 1962, Clarendon Press, Oxford, | MR

[37] Weyl, H.: Über gewöhnlicke Differentialgleichungen mit Singuritaten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Annal., 68, 1910, 220-269, | DOI | MR

[38] Yosida, K.: On Titchmarsh-Kodaira formula concerning Weyl-Stone eingenfunction expansion. Nagoya Math. J., 1, 1950, 49-58, | DOI | MR

[39] Yosida, K.: Lectures on Differential and Integral Equations. 1960, Springer, New York, | MR