Bulletin of the American Mathematical Society

MathSciNet review: 1567111
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: George E. Andrews
Title: The theory of partitions
Additional book information: in Encyclopedia of Mathematics and its Applications, volume 2, Addison-Wesley Publishing Company, Advanced Book Program, London, Amsterdam, Don Mills, Ontario, Sydney, and Tokyo, 1976, xiv + 255 pp., $19.50.

George E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 4082–4085. MR 351985, DOI 10.1073/pnas.71.10.4082

George E. Andrews, Problems and prospects for basic hypergeometric functions, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 191–224. MR 0399528

George E. Andrews, Partitions, $q$-series and the Lusztig-Macdonald-Wall conjectures, Invent. Math. 41 (1977), no. 1, 91–102. MR 446991, DOI 10.1007/BF01390165

4.

G. Andrews, An introduction to Ramanujan's "lost notebook", Amer. Math. Monthly (to appear).

George E. Andrews, Connection coefficient problems and partitions, Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978) Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979, pp. 1–24. MR 525316

George E. Andrews and Richard Askey, Enumeration of partitions: the role of Eulerian series and $q$-orthogonal polynomials, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 31, Reidel, Dordrecht-Boston, Mass., 1977, pp. 3–26. MR 519776

7.

P. Appell and E. Lacour, Principes de la théories desfonctions elliptiques, Gauthier-Villars, Paris, 1897.

Dennis Stanton and Mourad E. H. Ismail (eds.), {Dedicated to Richard A. Askey on the occasion of his 65th birthday. Part I}, International Press, Somerville, MA, 1999. Methods Appl. Anal. 6 (1999), no. 1. MR 1801153

Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR 0125252, DOI 10.1017/s0025557200044491

L. C. Biedenharn and J. D. Louck, Angular momentum in quantum physics, Encyclopedia of Mathematics and its Applications, vol. 8, Addison-Wesley Publishing Co., Reading, Mass., 1981. Theory and application; With a foreword by Peter A. Carruthers. MR 635121

D. M. Bressoud, A new family of partition identities, Pacific J. Math. 77 (1978), no. 1, 71–74. MR 506017

12.

D. Bressoud, Combinatorial proof of Schur's theorem (submitted).

13.

D. Bressoud, Extension of the partition sieve (submitted).

David M. Bressoud, A generalization of the Rogers-Ramanujan identities for all moduli, J. Combin. Theory Ser. A 27 (1979), no. 1, 64–68. MR 541344, DOI 10.1016/0097-3165(79)90008-6

15.

D. Bressoud, A functional generalization of the Rogers-Ramanujan identities with interpretation (submitted).

16.

A. Cauchy, Second Mémoire sur les fonctions dont plusieurs valeurs sont liées entre par une équation linéaire, Oeuvres, 1^re Série, Tome VIII, 50-55, Gauthier-Villars, Paris, 1893, reprinted from C. R. 1843.

Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258

18.

J. Dougall, A theorem of Sonine in Bessel functions, with two extensions to spherical harmonics, Proc. Edinburgh Math. Soc. 37 (1919), 33-47.

Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140–156. MR 143556, DOI 10.1063/1.1703773

20.

C. F. Gauss, Disquisitiones generales circa seriem infinitum 1 + αβx/1·γ + α(α + 1)β(β + 1)xx/1·2γ(γ + 1) + ..., Werke, vol. 3, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1866, pp. 123-162 (originally published in 1813).

21.

C. F. Gauss, Zur Theorie der neuen Transscendenten. II, Werke, vol. 3, Göttingen, 1866, pp. 436-445.

22.

C. F. Gauss, Hundert Theoreme über die neuen Transscendenten, Werke, vol. 3, Göttingen, 1866, pp. 461-469.

Basil Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math. 83 (1961), 393–399. MR 123484, DOI 10.2307/2372962

R. Wm. Gosper Jr., A calculus of series rearrangements, Algorithms and complexity (Proc. Sympos., Carnegie-Mellon Univ., Pittsburgh, Pa., 1976) Academic Press, New York, 1976, pp. 121–151. MR 0451617

G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1940. MR 0004860

26.

E. Heine, Untersuchungen über die Reihe $1+\frac{(1-q^\alpha)(1-q^\beta)}{(1-q)(1-q^\gamma)}\cdot x+\frac{(1-q^\alpha)(1+q^{\alpha}-1)(1-q^\beta)(1+q^{\beta}-1)}{(1-q)(1-q^2)(1-q^\gamma)(1+q^{\gamma}-1)}\cdot x^2+łdots$ J. Reine Angew Math. 34 (1845), 285-328.

27.

E. Heine, Theorie der Kugelfunctionen und der verwandten Functionen, Reimer, Berlin, 1878.

Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR 0453984

29.

C. Hermite, Oeuvres, Tome II, Gauthier-Villars, Paris, 1908, pp. 153-156; reprinted from Calcul différentiel et Calcul intégral de Lacroix, 6^e edition, Paris, 1862.

Egil A. Hylleraas, Linearization of products of Jacobi polynomials, Math. Scand. 10 (1962), 189–200. MR 145123, DOI 10.7146/math.scand.a-10527

31.

C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomontis, fratrum Bomtraeger 1829; reprinted in Gesammelte Werke, Volume 1, Reimer, Berlin, 1881; reprinted by Chelsea, New York, 1969, pp. 49-239.

I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931

G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Die Grundlehren der mathematischen Wissenschaften, Band 216, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR 0396134

34.

G. Racah, Theory of complex spectra. I, II, III, IV, Phys. Rev. 61 (1942), 186-197; 62 (1942), 438-462; 63 (1943), 367-382; 76 (1949), 1352-1365; reprinted in Quantum theory of angular momentum (L. Biedenharn and H. VanDam, ed.), Academic Press, New York, 1965.

34a. O. Rausenberger, Lehrbuch der Theorie der Periodischen Functionen einer Variabeln, Teubner, Leipzig, 1884.

35.

L. J. Rogers, On a three-fold symmetry in the elements of Heine's series, Proc. London Math. Soc. 24 (1893), 171-179.

36.

L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893), 337-352.

37.

L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.

38.

L. J. Rogers, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895), 15-32.

39.

K. H. Schellbach, Die Lehre von den Elliptischen Integralen und den Theta-Functionen, Reimer, Berlin, 1864.

40.

D. Stanton, Some basic hypergeometric polynomials arising from finite classical groups, Ph.D. thesis, Univ. of Wisconsin, Madison, 1977.

41.

H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms (II), Modular Functions in One Variable V (J.-P. Serre and D. B. Zagier, eds.), Springer-Verlag, Berlin and New York, 1977. MR 406931

H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55. MR 0406931

42. J. Wilson, Ph.D. thesis, Univ. of Wisconsin, Madison, 1978.

1.

G. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci U.S.A. 71 (1974), 4082-4085. MR 0351985

2.

G. Andrews, Problems and prospects for basic hypergeometric functions, Theory and Application of Special Functions (R. Askey, ed.), Academic Press, New York, 1975, pp. 191-224. MR 399528

3.

G. Andrews, q-series and the Lusztig-Macdonald-Wall conjectures, Invent Math. 41 (1977), 91-102. MR 446991

4.

G. Andrews, An introduction to Ramanujan's "lost notebook", Amer. Math. Monthly (to appear).

5.

G. Andrews, Connection coefficient problems and partitions, Proc. Sympos. Pure Math., vol. 34, Amer. Math. Soc. Providence, R. I., 1978, pp. 1-24. MR 525316

6.

G. Andrews and R. Askey, Enumeration of partitions: The role of Eulerian series and q-orthogonal polynomials Higher Combinatorics (M, Aigner, ed.), Reidel, Dordrecht, Holland, 1977, pp. 3-26. MR 519776

7.

P. Appell and E. Lacour, Principes de la théories desfonctions elliptiques, Gauthier-Villars, Paris, 1897.

8.

R. Askey and M. E.-H. Ismail, A generalization of ultraspherical polynomials (to appear in a book dedicated to P. Turán's memory). MR 1801153

9.

R. Bellman, A brief introduction to theta functions, Holt, Rinehart and Winston, New York, 1961. MR 125252

10.

L. Biedenharn and J. Louck, Angular momentum in quantum physics-Theory and application (to appear). MR 635121

11.

D. Bressoud, A new family of partition identities, Pacific J. Math. 77 (1978). MR 506017

12.

D. Bressoud, Combinatorial proof of Schur's theorem (submitted).

13.

D. Bressoud, Extension of the partition sieve (submitted).

14.

D. Bressoud, A generalization of the Rogers-Ramanujan identities for all moduli, J. Combinatorial Theory Ser. A (to appear). MR 541344

15.

D. Bressoud, A functional generalization of the Rogers-Ramanujan identities with interpretation (submitted).

16.

A. Cauchy, Second Mémoire sur les fonctions dont plusieurs valeurs sont liées entre par une équation linéaire, Oeuvres, 1^re Série, Tome VIII, 50-55, Gauthier-Villars, Paris, 1893, reprinted from C. R. 1843.

17.

P. Deligne, La conjecture de Weil. I, Inst Hautes Études Sci Publ. Math. 43 (1974), 273-307. MR 49 #5013. MR 340258

18.

J. Dougall, A theorem of Sonine in Bessel functions, with two extensions to spherical harmonics, Proc. Edinburgh Math. Soc. 37 (1919), 33-47.

19.

F. Dyson, Statistical theory of the energy level of complex systems. I, J. Math. Phys. 3 (1962), 140-156. MR 143556

20.

C. F. Gauss, Disquisitiones generales circa seriem infinitum 1 + αβx/1·γ + α(α + 1)β(β + 1)xx/1·2γ(γ + 1) + ..., Werke, vol. 3, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1866, pp. 123-162 (originally published in 1813).

21.

C. F. Gauss, Zur Theorie der neuen Transscendenten. II, Werke, vol. 3, Göttingen, 1866, pp. 436-445.

22.

C. F. Gauss, Hundert Theoreme über die neuen Transscendenten, Werke, vol. 3, Göttingen, 1866, pp. 461-469.

23.

B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math. 83 (1961), 393-399. MR 123484

24.

R. W. Gosper, A calculus of series rearrangements, Algorithms and Complexity, New Directions and Recent Results (J. Traub, ed.), Academic Press, New York, 1976, pp. 121-151. MR 451617

25.

G. H. Hardy, Ramanujan, Cambridge Univ. Press, London, 1940; reprinted by Chelsea, New York, 1959. MR 4860

26.

E. Heine, Untersuchungen über die Reihe $1+\frac{(1-q^\alpha)(1-q^\beta)}{(1-q)(1-q^\gamma)}\cdot x+\frac{(1-q^\alpha)(1+q^{\alpha}-1)(1-q^\beta)(1+q^{\beta}-1)}{(1-q)(1-q^2)(1-q^\gamma)(1+q^{\gamma}-1)}\cdot x^2+łdots$ J. Reine Angew Math. 34 (1845), 285-328.

27.

E. Heine, Theorie der Kugelfunctionen und der verwandten Functionen, Reimer, Berlin, 1878.

28.

P. Henrici, Applied and computational complex analysis, vol. 2, Wiley-Interscience, New York, 1977. MR 453984

29.

C. Hermite, Oeuvres, Tome II, Gauthier-Villars, Paris, 1908, pp. 153-156; reprinted from Calcul différentiel et Calcul intégral de Lacroix, 6^e edition, Paris, 1862.

30.

E. Hylleraas, Linearization of products of Jacobi polynomials, Math. Scand. 10 (1962), 189-200. MR 145123

31.

C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomontis, fratrum Bomtraeger 1829; reprinted in Gesammelte Werke, Volume 1, Reimer, Berlin, 1881; reprinted by Chelsea, New York, 1969, pp. 49-239.

32.

I. G. Macdonald, Affine root systems and Dedekind's η-function, Invent. Math. 15 (1972), 91-143. MR 357528

33.

G. Pólya and G. Szegö, Problems and theorems in analysis, vol. I, Springer-Verlag, New York, Heidelberg, Berlin, 1972. MR 396134

34.

G. Racah, Theory of complex spectra. I, II, III, IV, Phys. Rev. 61 (1942), 186-197; 62 (1942), 438-462; 63 (1943), 367-382; 76 (1949), 1352-1365; reprinted in Quantum theory of angular momentum (L. Biedenharn and H. VanDam, ed.), Academic Press, New York, 1965.

34a. O. Rausenberger, Lehrbuch der Theorie der Periodischen Functionen einer Variabeln, Teubner, Leipzig, 1884.




35.

L. J. Rogers, On a three-fold symmetry in the elements of Heine's series, Proc. London Math. Soc. 24 (1893), 171-179.

36.

L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893), 337-352.

37.

L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.

38.

L. J. Rogers, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895), 15-32.

39.

K. H. Schellbach, Die Lehre von den Elliptischen Integralen und den Theta-Functionen, Reimer, Berlin, 1864.

40.

D. Stanton, Some basic hypergeometric polynomials arising from finite classical groups, Ph.D. thesis, Univ. of Wisconsin, Madison, 1977.

41.

H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms (II), Modular Functions in One Variable V (J.-P. Serre and D. B. Zagier, eds.), Springer-Verlag, Berlin and New York, 1977. MR 406931

42.

J. Wilson, Ph.D. thesis, Univ. of Wisconsin, Madison, 1978.

42. J. Wilson, Ph.D. thesis, Univ. of Wisconsin, Madison, 1978.