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Spectral theory of singular Hahn difference equation of the Sturm-Liouville type. (English) Zbl 1476.39005

The authors study the spectral theory of singular Hahn difference equations of Sturm-Liouville type. They define a new Hilbert space and construct on it the Fourier transform. Then, they prove the Parseval equality and construct an expansion formula on a semi-unbounded interval.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
39A12 Discrete version of topics in analysis
39A70 Difference operators
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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[1] Aldwoah K.A.: Generalized time scales and associated difference equations. Ph.D. Thesis (2009)
[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. · Zbl 1424.39018
[3] Allahverdiev B.P., Tuna H.: Spectral expansion for singular Dirac system with impulsive conditions. Turkish J. Math. 42 (5) (2018) 2527-2545. · Zbl 1424.34307
[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. · Zbl 1438.34343
[5] Allahverdiev B.P., Tuna H.: The spectral expansion for Hahn-Dirac system on the whole line. Turkish J. Math. 43 (2019) 1668-1687. · Zbl 1468.34117
[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. · Zbl 1406.34101
[7] Allahverdiev B.P., Tuna H.: The Parseval equality and expansion formula for singular Hahn-Dirac system. In: Alparslan-Gok Z.S.: Emerging Applications of Differential Equations and Game Theory. IGI Global (2020) 209-235.
[8] Álvarez-Nodarse R.: On characterizations of classical polynomials. J. Comput. Appl. Math. 196 (1) (2006) 320-337. · Zbl 1108.33008
[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. · Zbl 1266.47054
[10] Annaby M.H., Hamza A.E., Makharesh S.D.: A Sturm-Liouville theory for Hahn difference operator. In: Xin Li, Zuhair Nashed: Frontiers of Orthogonal Polynomials and q-Series. World Scientific, Singapore (2018) 35-84. · Zbl 1415.39014
[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. · Zbl 1441.94061
[12] J. Arvesú : On some properties of q−Hahn multiple orthogonal polynomials. J. Comput. Appl. Math. 233 (6) (2010) 1462-1469. doi:10.1016/j.cam.2009.02.062 · Zbl 1179.33015
[13] J.M. Berezanskii: Expansions in Eigenfunctions of Selfadjoint Operators. Amer. Math. Soc., Providence (1968). · Zbl 0157.16601
[14] A. Dobrogowska, A. Odzijewicz: Second order q-difference equations solvable by factorization method. J. Comput. Appl. Math. 193 (1) (2006) 319-346. · Zbl 1119.39017
[15] G.Sh. Guseinov: Eigenfunction expansions for a Sturm-Liouville problem on time scales. Int. J. Difference Equat. 2 (1) (2007) 93-104. · Zbl 1145.39005
[16] G.Sh. Guseinov: An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3 (1) (2008) 147-160.
[17] W. Hahn: Über orthogonalpolynome, die q-Differenzengleichungen genügen. Math. Nachr. 2 (1949) 4-34. · Zbl 0031.39001
[18] W. Hahn: Ein Beitrag zur Theorie der Orthogonalpolynome. Monatsh. Math. 95 (1983) 19-24. · Zbl 0497.33009
[19] A.E. Hamza, S.A. Ahmed: Existence and uniqueness of solutions of Hahn difference equations. Adv. Difference Equat. 316 (2013) 1-15. · Zbl 1391.39015
[20] A.E. Hamza, S.D. Makharesh: Leibniz’ rule and Fubinis theorem associated with Hahn difference operator. J. Adv. Math. 12 (6) (2016) 6335-6345.
[21] A. Huseynov, E. Bairamov: On expansions in eigenfunctions for second order dynamic equations on time scales. Nonlinear Dyn. Syst. Theory 9 (1) (2009) 77-88. · Zbl 1176.34107
[22] A. Huseynov: Eigenfunction expansion associated with the one-dimensional Schrödinger equation on semi-infinite time scale intervals. Rep. Math. Phys. 66 (2) (2010) 207-235. · Zbl 1226.34088
[23] F.H. Jackson: q-Difference equations. Amer. J. Math. 32 (1910) 305-314. · JFM 41.0502.01
[24] D.L. Jagerman: Difference Equations with Applications to Queues. Dekker, New York (2000). · Zbl 0963.39001
[25] C. Jordan: Calculus of Finite Differences, 3rd edn. Chelsea, New York (1965). · Zbl 0154.33901
[26] A.N. Kolmogorov, S.V. Fomin: Introductory Real Analysis. Translated by R.A. Silverman. Dover Publications, New York (1970).
[27] K.H. Kwon, D.W. Lee, S.B. Park, B.H. Yoo: Hahn class orthogonal polynomials. Kyungpook Math. J. 38 (1998) 259-281. · Zbl 0922.33005
[28] P.A. Lesky: Eine Charakterisierung der klassischen kontinuierlichen, diskretenund q-Orthgonalpolynome. Shaker, Aachen (2005).
[29] N. Levinson: A simplified proof of the expansion theorem for singular second order linear differential equations. Duke Math. J. 18 (1951) 57-71. · Zbl 0044.31302
[30] B.M. Levitan, I.S. Sargsjan: Sturm-Liouville and Dirac Operators. Springer (1991).
[31] M.A. Naimark: Linear Differential Operators, 2nd edn., 1968. Nauka, Moscow (1969). English translation of 1st edn. · Zbl 0227.34020
[32] J. Petronilho: Generic formulas for the values at the singular points of some special monic classical H_q,ω -orthogonal polynomials.. J. Comput. Appl. Math. 205 (2007) 314-324. · Zbl 1119.33010
[33] T. Sitthiwirattham: On a nonlocal boundary value problem for nonlinear second-order Hahn difference equation with two different q,ω -derivatives. Adv. Difference Equat. 2016 (1) (2016). Article number 116 · Zbl 1419.39010
[34] M.H. Stone: A comparison of the series of Fourier and Birkho. Trans. Amer. Math. Soc. 28 (1926) 695-761. · JFM 52.0456.02
[35] M.H. Stone: Linear Transformations in Hilbert Space and Their Application to Analysis. Amer. Math. Soc. (1932). · Zbl 0005.40003
[36] E.C. Titchmarsh: Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition. Clarendon Press, Oxford (1962). · Zbl 0099.05201
[37] H. Weyl: Über gewöhnlicke Differentialgleichungen mit Singuritaten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Annal. 68 (1910) 220-269. · JFM 41.0343.01
[38] K. Yosida: On Titchmarsh-Kodaira formula concerning Weyl-Stone eingenfunction expansion. Nagoya Math. J. 1 (1950) 49-58. · Zbl 0038.24802
[39] K. Yosida: Lectures on Differential and Integral Equations. Springer, New York (1960). · Zbl 0090.08401
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