×

zbMATH — the first resource for mathematics

Rényi entropies, \(L_q\) norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions. (English) Zbl 1329.33013
Summary: The quantification of the spreading of the orthogonal polynomials \(p_n(x)\) can be investigated by means of the Rényi entropies \(R_q[{\rho}]\), \(q\) being a positive integer number, of the associated Rakhmanov probability densities, \({\rho}(x)={\omega}(x)p_n^2(x)\), where \({\omega}(x)\) is the corresponding weight function. The Rényi entropies are closely related to the \(L_q\)-norms of the polynomials. In this manuscript, the \(L_q\)-norms and the associated Rényi entropies of the real hypergeometric orthogonal polynomials (i.e., Hermite, Laguerre, and Jacobi polynomials) and the generalized Hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of Lauricella and Srivastava-Daoust types which are evaluated at some specific values of \(2q\) variables. These functions depend on \(4q+1\) and \(6q+2\) parameters, respectively, which are determined by the order \(q\), the degree \( n\) of the polynomial, and the parameters of the orthogonality weight function \({\omega}(x)\). The key idea is based on some extended linearization formulas for these polynomials. These results open the way to determine the Rényi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials.

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Álvarez-Nodarse, R.; Yáñez, R. J.; Dehesa, J. S., Modified clebsh-Gordan-type expansions for products of discrete hypereometric polynomials, J. Comput. Appl. Math., 89, 171-197, (1997) · Zbl 0909.33006
[2] Andrews, G. E.; Askey, R.; Roy, R., Special functions, (Encyclopedia for Mathematics and its Applications, (1999), Cambridge University Press)
[3] Angulo, J. C.; Romera, E.; Dehesa, J. S., Inverse atomic densities and inequalities among density functionals, J. Math. Phys., 41, 7906-7917, (2000) · Zbl 0970.81101
[4] Anshelevich, M., Linearization coefficients for orthogonal polynomials using stochastic processes, Ann. Prob., 33, 114-136, (2005) · Zbl 1092.05076
[5] Aptekarev, A. I.; Buyarov, V. S.; Dehesa, J. S., Asymptotic behavior of the \(L^p\)-norms and the entropy for general orthogonal polynomials, Russian Acad. Sci. Sb. Math., 82, 373-395, (1995)
[6] Aptekarev, A. I.; Dehesa, J. S.; Sánchez-Moreno, P.; Tulyakov, N., Asymptotics of \(L_p\) norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states, Contemp. Math., 578, 19-29, (2012) · Zbl 1318.94027
[7] Aptekarev, A. I.; Martínez-Finkelshtein, A.; Dehesa, J. S., Asymptotics of orthogonal polynomial’s entropy, J. Comput. Appl. Math., 233, 1355-1365, (2010) · Zbl 1183.94013
[8] Area, I.; Godoy, E.; Ronveaux, A.; Zarzo, A., Solving connection and linearisation problems within the Askey scheme and its q-analogue via inversion formulas, J. Comput. Appl. Math., 133, 151-162, (2001) · Zbl 0988.33008
[9] Artes, P. L.; Dehesa, J. S.; Martínez-Finkelstein, A.; Sánchez-Ruiz, J., Linearisation and connection coefficients for hypergeometric polynomials, J. Comput. Appl. Math., 99, 15-26, (1998) · Zbl 0927.33005
[10] R.A. Askey, Orthogonal polynomials and special functions, in: CBMS-NSF Reg. Conferences Series in Applied Mathematics, vol. 21, SIAM, Philadelphia, Pennsylvania.
[11] Askey, R. A.; Gasper, G., Linearization of the product of Jacobi polynomials III, Can. J. Math., 23, 332-338, (1971) · Zbl 0212.40904
[12] Azor, R.; Gillis, J.; Victor, J. D., Combinatorial applications of Hermite polynomials, SIAM J. Math. Anal., 13, 879-890, (1982) · Zbl 0516.33008
[13] Bashkirov, A. G., Maximum renyi entropy principle for systems with power-law Hamiltonians, Phys. Rev. Lett., 93, 130601, (2004)
[14] Belmehdi, S.; Lewanowicz, S.; Ronveaux, A., Linearization of products of orthogonal polynomials of a discrete variable, Appl. Math., 24, 445-455, (1997) · Zbl 0891.33006
[15] Bernstein, S. N., Complete Works, vol. 2, (1954), Ac. Sci. USSR Publ. Moscow
[16] Brody, D. C.; Buckley, I. R.C.; Constantinou, I. C., Option price calibration from Rényi entropy, Phys. Lett. A, 366, 298-307, (2007) · Zbl 1203.91284
[17] Busbridge, I. W., Some integrals involving Hermite polynomials, J. Lond. Math. Soc., 23, 135-141, (1948) · Zbl 0032.27601
[18] Carlitz, L., Some integrals containing products of Legendre polynomials, Arch. Math., 12, 334-340, (1961) · Zbl 0113.27802
[19] Chihara, T. S., An introduction to orthogonal polynomials, (1978), Gordon and Breach New York · Zbl 0389.33008
[20] Daalhuis, A. B.O., Uniform asymptotic expansions for hypergeometric functions with large parameters II, Anal. Appl., 1, 121-128, (2003) · Zbl 1048.33005
[21] J.S. Dehesa, A. Guerrero, J.L. López, P. Sánchez-Moreno, Asymptotics (\(p \rightarrow \infty\)) of \(L_p\)-norms of hypergeometric ortogonal polynomials, J. Math. Chem. (2013), accepted for publication.
[22] Dehesa, J. S.; Guerrero, A.; Sánchez-Moreno, P., Information-theoretic-based spreading measures of ortogonal polynomials, Complex Anal. Oper. Theory, 6, 585-601, (2012) · Zbl 1276.33011
[23] Dehesa, J. S.; López-Rosa, S.; Manzano, D., Entropy and complexity analyses of \(D\)-dimensional quantum systems, (Sen, K. D., Statistical Complexities: Application to Electronic Structure, (2012), Springer Berlin)
[24] Dehesa, J. S.; Martínez-Finkelshtein, A.; Sánchez-Ruiz, J., Quantum information entropies and orthogonal polynomials, J. Comput. Appl. Math., 133, 23-46, (2001) · Zbl 1008.81014
[25] Erdélyi, A., On some expansions in Laguerre polynomials, J. London Math. Soc., 13, 154-156, (1938) · JFM 64.0355.02
[26] Even, S.; Gillis, J., Derangements and Laguerre polynomials, Math. Proc. Camb. Phil. Soc., 79, 135-143, (1976) · Zbl 0325.05006
[27] Feldheim, E., Relations entre LES polynômes de Jacobi, Laguerre et Hermite, Acta Math., 74, 117-138, (1941) · JFM 68.0152.04
[28] Ferreira, C.; López, J. L.; Pérez-Sinusia, E., The Gauss hypergeometric function \(F(a, b \text{;} c \text{;} z)\) for large c, J. Comput. Appl. Math., 197, 568-577, (2006) · Zbl 1106.33004
[29] Foata, D.; Strehl, V., Combinatorics of Laguerre polynomials, enumeration and design, (1982), Waterloo Ontario
[30] Foata, D.; Zeilberger, D., Laguerre polynomials, weighted derangements, and positivity, SIAM J. Discr. Math., 1, 425-433, (1988) · Zbl 0662.05003
[31] A. Guerrero, P. Sánchez-Moreno, J.S. Dehesa, \(L_q\)-norm asymptotics of generalised Hermite polynomials (2013), in press.
[32] Guerrero, A.; Sánchez-Moreno, P.; Dehesa, J. S., Information-theoretic lengths of Jacobi polynomials, J. Phys. A: Math. Theor., 43, 305203, (2010) · Zbl 1220.33009
[33] Hounkonnou, M. N.; Belmehdi, S.; Ronveaux, A., Linearization of arbitrary products of classical orthogonal polynomials, Appl. Math., 27, 187-196, (2000) · Zbl 0997.33002
[34] Hylleraas, E., Linearization of products of Jacobi polynomials, Math. Scand., 10, 189-200, (1962) · Zbl 0109.29603
[35] M.E.H. Ismail, Classical and quantum orthogonal polynomials in one variable. with two chapters by Walter Van Assche. With a foreword by Richard A. Askey. Reprint of the 2005 original. Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. · Zbl 1172.42008
[36] Ismail, M. E.H.; Kasraoui, A.; Zeng, J., Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials, J. Combinat. Theory, 120, 561-599, (2013) · Zbl 1259.05022
[37] M.E.H. Ismail, P. Simeonov, Asymptotics of generalised derangements, Adv. Comput. Math. (2011). accepted for publication. D.O.I.: 10.1007/s10444-011-9271-7.
[38] Jizba, P.; Arimitsu, T., The world according to Rényi: thermodynamics of multifractal systems, Ann. Phys., 312, 17-59, (2004) · Zbl 1044.82001
[39] Kim, D.; Zeng, J., A combinatorial formula for the linearization coefficients of general Sheffer polynomials, Eur. J. Combinat., 22, 313-332, (2001) · Zbl 0977.05144
[40] Koekoek, R.; Lesky, P. A.; Swarttouw, R. F., Hypergeometric orthogonal polynomials and their q-analogues, with a foreword by tom H. Koornwinder, (2010), Springer Monographs in Mathematics, Springer-Verlag Berlin · Zbl 1200.33012
[41] M.G. Krein, A.A. Nudel’man, The Markov Moment Problem and Extremal Problems, in: Translations of Mathematical Monographs, volume 50, AMS, Providence, 1977.
[42] Labelle, J.; Yeh, Y. N., The combinatorics of Laguerre, Charlier and Hermite polynomials, Studies in Appl. Math., 80, 25-36, (1980) · Zbl 0668.33006
[43] Larsson-Cohn, L., \(L^p\)-norms of Hermite polynomials and an extremal problem on Wiener chaos, Arkiv Mat., 40, 133-144, (2002) · Zbl 1021.60043
[44] Lasser, R., Linearization of the product of associated Legendre polynomials, SIAM J. Math. Anal., 14, 403-408, (1983) · Zbl 0509.33007
[45] Leo, P. A.; Ong, S. H.; Srivastava, H. M., Some integrals of the products of Laguerre polynomials, Internat. J. Comput. Math., 78, 303-321, (2001) · Zbl 1018.33009
[46] Lewanowicz, S., Second order recurrence relations for the linearization coefficients of the classical orthogonal polynomials, J. Comput. Appl. Math., 69, 159-170, (1996) · Zbl 0885.33003
[47] Liu, S.; Parr, R. G., Expansions of the correlation energy density functional and its kinetic-energy component in terms of homogeneous functionals, Phys. Rev. A, 53, 2211, (1996)
[48] Liu, S.; Parr, R. G., Expansions of density functionals: justification and nonlocal representation of the kinetic energy, exchange energy, and classical Coulomb repulsion energy for atoms, Physica A, 55, 1792, (1997)
[49] D.S. Lubinsky, E.B. Saff, Strong asymptotics for extremal polynomials associated with weights on \(\mathbb{R}\), in: Lecture Notes in Mathematics, volume 1305, Springer-Verlag, Belin, 1988. · Zbl 0647.41001
[50] Nagy, A.; Liu, S.; Parr, R. G., Density-functional formulas for atomic electronic energy components in terms of moments of the electron density, Phys. Rev. A, 59, 3349, (1999)
[51] Nikiforov, A. F.; Uvarov, V. B., Special functions in mathematical physics, (1988), Birkäuser-Verlag Basel · Zbl 0624.33001
[52] Niukkanen, A. W., Clebsch-Gordan-type linearisation relations for the products of Laguerre polynomials and hydrogen-like functions, J. Phys. A: Math. Gen., 18, 1399-1417, (1985) · Zbl 0582.33008
[53] Piquette, J. C., Applications of a technique for evaluating indefinite integrals containing products of the special functions of physics, SIAM J. Math. Anal., 20, 1260-1269, (1989) · Zbl 0674.33004
[54] Prudnikov, A. P.; Brychkov, Y. A.; Marichev, O. I., Integrals and series, (1986), Gordon and Breach Science Publishers Amsterdam · Zbl 0606.33001
[55] Rahman, M., A non-negative representation of the linearization coefficients of the product of Jacobi polynomials, Can. J. Math., 33, 915-928, (1981) · Zbl 0423.33003
[56] Rakhmanov, E. A., On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb., 32, 199-213, (1977) · Zbl 0401.30033
[57] E.A. Rakhmanov, On a conjecture of V. A. Steklov in the theory of orthogonal polynomials, Math. USSR Sb. 36 (1980) 549-575. · Zbl 0452.33012
[58] Rényi, A., Probability theory, (1970), North Holland Amsterdam · Zbl 0206.18002
[59] Romera, E.; Angulo, J. C.; Dehesa, J. S., The Hausdorff entropic moment problem, J. Math. Phys., 42, 2309-2314, (2001) · Zbl 1009.44003
[60] E. Romera, J.C. Angulo, J.S. Dehesa, Reconstruction of a density from its entropic moments, in: R.L. Fry (Ed.), Proc. 21st Int. Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, American Institute of Physics, New York, 2002, p. 449.
[61] Ronveaux, A.; Hounkonnou, M. N.; Belmehdi, S., Generalised linearization problems, J. Phys. A: Math. Gen., 28, 4423-4430, (1995) · Zbl 0867.33003
[62] de Sainte-Catherine, M.; Viennot, G., Combinatorial interpretation of integrals of products of Hermite, Laguerre and Tchebycheff polynomials, Lect. Notes in Math., 1171, 120-128, (1985) · Zbl 0587.05003
[63] Sánchez-Moreno, P.; Dehesa, J. S.; Manzano, D.; Yáñez, R. J., Spreading lengths of Hermite polynomials, J. Comput. Appl. Math., 233, 2136-2148, (2010) · Zbl 1188.33017
[64] Sánchez-Moreno, P.; Manzano, D.; Dehesa, J. S., Direct spreading measures of Laguerre polynomials, J. Comput. Appl. Math., 235, 1129-1140, (2011) · Zbl 1223.33017
[65] Sánchez-Ruiz, J.; Artes, P. L.; Martínez-Finkelstein, A.; Dehesa, J. S., General linearisation relations for products of continuous hypergeometric-type polynomials, J. Phys. A: Math. Gen., 32, 7345-7366, (1999) · Zbl 0945.33006
[66] Sen, K. D., Statistical Complexity: Applications in Electronic Structure, (2011), Springer Heidelberg
[67] Slater, L. J., Generalized hypergeometric functions, (1966), Cambridge University Press Cambridge · Zbl 0135.28101
[68] Srivastava, H. M., A unified theory of polynomial expansions and their applications involving Clebsch-Gordan type linearization relations and Neumann series, Astrophys. Space Sci., 150, 251-266, (1988) · Zbl 0644.33006
[69] Srivastava, H. M.; Karlsson, P. W., Multiple Gaussian hypergeometric series, (1985), John Wiley and Sons New York · Zbl 0552.33001
[70] Srivastava, H. M.; Mavromatis, H. A.; Alassar, R. S., Remarks on some associated Laguerre integral results, Appl. Math. Lett., 16, 1131-1136, (2003) · Zbl 1058.33012
[71] Srivastava, H. M.; Niukkanen, A. W., Some Clebsch-Gordan type linearization relations and associated families of Dirichlet integrals, Math. Comput. Model., 37, 245-250, (2003) · Zbl 1076.33008
[72] Suetin, P. K., A problem of V.A. Steklov in the theory of orthogonal polynomials, J. Soviet Math., 12, 631-682, (1979) · Zbl 0473.42016
[73] G. Szegö, Orthogonal Polynomials, volume 23 of American Mathematical Society, Colloquium Publications, Amer. Math. Soc., Providence, 2003.
[74] Szwarc, R., Linearization and connection coefficients of orthogonal polynomials, Monatshefte Math., 113, 319-329, (1992) · Zbl 0766.33008
[75] Temme, N. M., Special functions: an introduction to the classical functions of mathematical physics, (1996), Wiley-Intersciente New York · Zbl 0856.33001
[76] Temme, N. M., Large parameter cases of the Gauss hypergeometric function, J. Comput. Appl. Math., 153, 441-462, (2003) · Zbl 1019.33003
[77] Tsallis, C., Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 52, 479-487, (1998) · Zbl 1082.82501
[78] Tsallis, C., Nonextensive statistics: theoretical, experimental and computational evidences and connections, Braz. J. Phys., 29, 1-35, (1999)
[79] Tsallis, C., Introduction to nonextensive statistical mechanics, (2009), Springer · Zbl 1172.82004
[80] Turan, P., Selected papers of alfréd Rényi, (1976), Akademia Kiado Budapest
[81] A. Zygmund, Trigonometric Series. With a foreword by Robert A. Fefferman, volume I - II of Cambridge Mathematical Library, third ed., Cambridge University Press, Cambridge, 2002. · Zbl 1084.42003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.