From: Wolfram Koepf (koepf@zib.de)

Subject: **Compiled Booklist**

Here I give the announced list of books and survey papers that should give a basis for understanding the current trends and needs in OP-SF. Let me say that the given list can be neither complete nor perfect. If somebody feels that an important item is missing, please let me know.

Readers who are interested in classical orthogonal polynomials should probably first look at [Chihara], [NU], [Rainville], [Szego], or [Tricomi]; readers looking for applications in mathematical physics might consider [NU].

[Artin] Artin, E.: "The Gamma Function." Holt, Rinehart and Winston, New York, 1964. [Askey1] Askey, R. A. (Ed.): "Theory and Application of Special Functions". Proceedings of an advanced seminar sponsored by the Mathematics Research Center, University of Wisconsin-Madison, March 21-April 2, 1975. Academic Press, New York, 1975. [Askey2] Askey, R. A. (Ed.): "Orthogonal Polynomials and Special Functions." Regional Conference Series in Applied Mathematics 21, SIAM, Philadelphia, 1975. [AKS] Askey, R. A., Koornwinder, T. H. and Schempp, W. (Eds.): "Special Functions: Group Theoretical Aspects and Applications." Mathematics and Its Applications, Vol. 18. Reidel, Dordrecht-Boston-Lancaster, 1984. [AW3] Askey, R. A. and Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc. 319, Providence, Rhode Island, 1985. [Bailey] Bailey, W. N.: "Generalized Hypergeometric Series". Cambridge University Press, England, 1935; reprinted 1964 by Stechert-Hafner Service Agency, New York-London. [Carlson] Carlson, B. C.: "Special Functions of Applied Mathematics." Academic Press, New York, 1977. [Chihara] Chihara, T. S.: "An Introduction to Orthogonal Polynomials." Gordon and Breach Publ., New York, 1978. [Freud] Freud, G.: "Orthogonale Polynome". Birkhauser, Basel, 1969; English translation, Pergamon Press, Oxford, 1971. [GR] Gasper, G. and Rahman, M.: "Basic Hypergeometric Series." Encyclopedia of Mathematics and Its Applications, Vol. 34. Cambridge University Press, 1990. [Geronimus1] Geronimus, Ya. L.: "Polynomials Orthogonal on a Circle and Interval." International Series of Monographs on Pure and Applied Mathematics, Vol. 18. Pergamon Press, Oxford-London-New York-Paris, 1961. [Geronimus2] Geronimus, Ya. L.: Appendix to the Russian translation of Szego's book "Orthogonal Polynomials." Staatsverlag fur physikalisch-mathematische Literatur, Moskau, 1962. [GHMM] Gottloeber, S., Haubold, H. J., Muecket, J.-P. and Mueller, V.: "Early Evolution of the Universe and Formation of Structure." Akademie-Verlag, Berlin, 1990. [Henrici1] Henrici, P.: "Applied and Computational Complex Analysis, Vol. 1: Power Series, Integration, Conformal Mapping, Location of Zeros." John Wiley & Sons, New York, 1974. [Henrici2] Henrici, P.: "Applied and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions." John Wiley & Sons, New York, 1977. [Henrici3] Henrici, P.: "Applied and Computational Complex Analysis, Vol. 3: Discrete Fourier Analysis - Cauchy Integrals - Construction of Conformal Maps- Univalent Functions." John Wiley & Sons, New York, 1986. [Hai-Yakubovich] Hai, Nguyen Thanh and Yakubovich, S. B.: "The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory", World Scientific. Series Soviet and East European Mathematics. Vol.6. 1992. 295 pp. [Hua] Hua, L.K.: "Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains." Translations of Mathematical Monographs, Vol. 6, Amer. Math. Soc., Providence, Rhode Island, 1963. [Lebedev] Lebedev, N. N.: "Special Functions and their Applications." Translated and edited by Richard A. Silverman. Dover Publications, New York, 1972. [LL] Levin, A. and Lubinsky, D. S.: "Christoffel Functions and Orthogonal Polynomials for Exponential Weights on [-1,1]." Memoirs of the American Mathematical Society, No.535, Vol. 11, 1994. [LW] Lorentzen, L. and Waadeland, H.: "Continued Fractions with Applications." Studies in Computational Mathematics, Vol. 3. North-Holland, Amsterdam, 1992. [Lubinsky] Lubinsky, D. S.: "Strong Asymptotics for Extremal Errors and Polynomials Associated with Erdos Type Weights."Pitman Research Notes in Mathematics, Vol. 202, Longman, Harlow, Essex, 1989. [LS] Lubinsky, D. S. and Saff, E. B.: "Strong Asymptotics for Extremal Polynomials Associated with Exponential Weights." Lecture Notes in Mathematics, Vol. 1305, Springer, Berlin, 1988. [Luke] Luke, Y.L.: "The Special Functions and their Approximations." Vols. 1 and 2, Academic Press, New York 1969. [MH] Mathai, A. M. and Haubold, H. J.: "Modern Problems in Nuclear and Neutrino Astrophysics." Akademie-Verlag, Berlin, 1988. [Miller1] Miller, W., Jr.: "Lie Theory and Special Functions." Academic Press, New York, 1968. [Miller2] Miller, W., Jr.: "Symmetry and Separation of Variables." Encyclopedia of Mathematics and Its Applications, Vol. 4. Addison-Wesley, Reading, Massachusetts, 1977. [Nevai1] Nevai, P. G.: "Orthogonal Polynomials." Memoirs Amer. Math. Soc., Vol. 213, Providence, Rhode Island, 1979. [Nevai2] Nevai, P. (Ed.): "Orthogonal Polynomials: Theory and Practice." Proceedings of the NATO Advanced Study Institute on Orthogonal Polynomials and Their Applications, Colombus, Ohio, U.S.A., May 22-June 3, 1989, Kluwer Academic Publ., Dordrecht-Boston-London, 1990. [NU] Nikiforov, A. F. and Uvarov, V. B.: "Special Functions of Mathematical Physics". Translated from the Russian by R. P. Boas, Birkhauser, Basel, 1988. [NSU] Nikiforov, A. F., Suslov, S. K. and Uvarov, V. B.: "Classical Orthogonal Polynomials of a Discrete Variable." Springer-Verlag, Berlin-Heidelberg-New York, 1991. [NS] Nikishin, E. M. and Sorokin, V. N.: "Rational Approximations and Orthogonality." Translations of Mathematical Monographs 92, Amer. Math. Soc., Providence, Rhode Island, 1991. [Olver] Olver, F. W. J.: "Asymptotics and Special Functions." Academic Press, New York, 1974. (reprinted A. K. Peters, Wellesley, 1997) [Perron] Perron, O.: "Die Lehre von den Kettenbruchen." Teubner, Leipzig, 1913; second edition, Chelsea, New York, 1950. [PWZ] Petkovvsek, M., Wilf, H. S. and Zeilberger, D.: "A=B." A. K. Peters, Wellesley, 1996. [Rainville] Rainville, E. D.: "Special Functions". The MacMillan Co., New York, 1960. [Richards] Richards, D. St. P. (Ed.): "Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications." Proceedings of an AMS special session, March 22-23, 1991 in Tampa, FL, USA. Contemporary Mathematics 138, Amer. Math. Soc., Providence, Rhode Island, 1992. [Ronveaux] Ronveaux, A. (Ed.): "Heun's Differential Equations." Oxford University Press, New York, 1995. [Shohat] Shohat, J. A. and Tamarkin, J. D. : "The Problem of Moments." Amer. Math. Soc., Providence, Rhode Island, 1963. [Szego] Szego, G.: "Orthogonal Polynomials." Amer. Math. Soc. Coll. Publ., Vol. 23, New York, 1939; 4th Edition, 1975. [ST] Stahl, H. and Totik, V.: "General Orthogonal Polynomials." Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 1992. [Talman] Talman, J.: "Special Functions: a Group Theoretic Approach." W. A. Benjamin, New York, 1968. [Temme] Temme, N. M.: "Special Functions. An Introduction to the Classical Functions of Mathematical Physics." John Wiley & Sons Inc., New York, 1996. [Totik] Totik, V.: "Weighted Approximation with Varying Weight." Springer Lecture Notes in Mathematics, Vol. 1569, Springer, Berlin, 1994. [Tricomi] Tricomi, F. G.: "Vorlesungen uber Orthogonalreihen." Grundlehren der Mathematischen Wissenschaften 76, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1955. [VanAssche] Van Assche, W.: "Asymptotics for Orthogonal Polynomials." Lecture Notes Math. 1265, Springer, Berlin-Heidelberg-New York, 1987. [Vilenkin1] Vilenkin, N. Ya.: "Special Functions and the Theory of Group Representations." Translations of Mathematical Monographs 22, Amer. Math. Soc., Providence, Rhode Island, 1968. [Vilenkin2] Vilenkin, N. Ya., and Klimyk, A. U.: "Representation of Lie Groups and Special Functions, Vols. 1-3, and "Recent Advances", Kluwer Academic Publ., Dordrecht-Boston-London, 1992-1995. [Wall] Wall, H. S.: "Analytic Theory of Continued Fractions." Chelsea, Bronx, NY, 1973. [Wong] Wong, R.: "Asymptotic Approximations of Integrals." Academic Press, New York, 1989. [Yakubovich-Luchko] Yakubovich, S. B. and Luchko, Yu. F.: "The Hypergeometric Approach to Integral Transforms and Convolutions". Kluwer Academic Publishers. Mathematics and Applications. Vol.287. 1994. 234 pp. [Yakubovich] Yakubovich, S. B.: "Index Transforms", World Scientific. 1996. 258 pp.

The following are handbooks and other reference manuals for orthogonal polynomials and special functions.

[AS] Abramowitz, M. and Stegun, I. A.: "Handbook of Mathematical Functions." Dover Publ., New York, 1964. [Erdelyi1] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: "Higher Transcendental Functions, Vols. 1-3." McGraw-Hill, New York, 1953-1955. [Erdelyi2] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: "Tables of Integral Transforms, Vols. 1-2." McGraw-Hill, New York, 1954. [GR] Gradshteyn, I. S. and Ryzhik, I. M.: "Table of Integrals, Series, and Products." Printed and CD-ROM Version. Academic Press, San Diego, California, 1996. [Han] Hansen, E. R.: "A Table of Series and Products." Prentice-Hall, Englewood Cliffs, NJ, 1975. [Swarttouw] Koekoek, R and Swarttouw, R.F. "The Askey scheme of hypergeometric orthogonal polynomials and its q-analogues," Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report 94-05, 1994; revised version of February 20, 1996 obtainable by anonymous ftp from: unvie6.un.or.at, directory siam/opsf_new/koekoek_swarttouw, file koekoek_swarttouw1.ps, or via World Wide Web: ftp://unvie6.un.or.at/siam/opsf_new/koekoek_swarttouw/koekoek_swarttouw1.ps hypertext version: http://www.can.nl/~demo/AWscheme/intro.html [MOS] Magnus, W., Oberhettinger, F. and Soni, R. P.: "Formulas and Theorems for the Special Functions of Mathematical Physics". Springer, Berlin-Heidelberg-New York, 1966. [Mathai] Mathai, A. M.: "A Handbook of Generalized Special Functions for Statistical and Physical Sciences." Clarendon Press, Oxford, 1993. [PBM] Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I.: "Integrals and Series, Vols.1-3." Gordon and Breach Publ., New York, 1989-1990.

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