Abstract
The theory of partitions has long been associated with so called basic hypergeometric functions or Eulerian series. We begin with discussion of some of the lesser known identities of L.J. Rogers which have interesting interpretations in the theory of partitions. Illustrations are given for the numerous ways partition studies lead to Eulerian series. The main portion of our work is primarily an introduction to recent work on orthogonal polynomials defined by basic hypergeometric series and to the applications that can be made of these results to the theory of partitions. Perhaps it is most interesting to note that we deduce the Rogers-Ramanujan identities from our solution to the connection coefficient problem for the little q-Jacobi polynomials.
Partially supported by the United States Army under Contract No. DAAG29-75-C-0024 and National Science Foundation Grant MSP74-07282.
Partially supported by the United States Army under Contract No. DAAG29-75-C-0024 and National Science Foundation Grant MCS75-06687-3.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G.E. Andrews, An analytic proof of the Rogers-Ramanujan-Gordon identities, Amer. J. Math., 88 (1966), 844–846.
G.E. Andrews, q-Identities of Auluk, Carlitz and Rogers, Duke Math. J., 33 (1966), 575–582.
G.E. Andrews, Number Theory, W.B. Saunders, Philadelphia, 1971.
G.E. Andrews, On the q-analog of Rummer’s theorem and applications, Duke Math. J., 40 (1973), 525–528.
G.E. Andrews, A general theory of identities of the Rogers-Ramanujan type, Bull. Amer. Math. Soc., 80 (1974), 1033–1052.
G.E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci., 71 (1974), 4082–4085.
G.E. Andrews, Problems and propsects for basic hypergeometric functions, from Theory and Application of Special Functions, ed. R. Askey, Academic Press, New York, 1975, pp. 191–224.
G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, Reading, 1977.
G.E. Andrews and R.A. Askey, The Classical and Discrete Orthogonal Polynomials and Their q-Analogs.
R.A. Askey, Orthogonal Polynomials and Special Functions, Regional Conf. Series in Appl. Math., Vol. 21, S.I.A.M., Philadephia, 1975.
W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935 (Reprinted: Hafner, New York, 1964 ).
W.N. Bailey, A note on certain q–identities, Quart. J. Math., 12 (1941), 173–175.
W.G. Connor, Partition theorems related to some identities of Rogers and Watson, Trans. Amer. Math. Soc., 214 (1975), 95–111.
J.A. Daum, The basic analog of Kummer’s theorem, Bull. Amer. Math. Soc., 48 (1942), 711–713.
B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math., 83 (1961), 393–399.
B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J., 32 (1965), 741–748.
G.H. Hardy, Ramanujan, Cambridge University Press, Cambridge, 1940 (Reprinted: Chelsea, New York).
G.H. Hardy and E.M. Wright, An Introduction to The Theory of Numbers, 4th ed., Oxford University Press, Oxford, 1960.
W. Hahn, Über Orthogonalpolynome, die q–Differenzengleichungen genügen, Math. Nach., 2 (1949), 4–34.
L.J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc., 25 (1894), 318–343.
L.J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc.(2), 16 (1917), 315–336.
W.A. Al-Salam and A. Verma, Orthogonality preserving operators and q-Jacobi polynomials, to appear.
L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
G. Szego, Orthogonal Polynomials, Colloquium Publications, vol. 23, 3rd ed., American Mathematical Society, Providence, 1967.
G.N. Watson, A new proof of the Rogers-Ramanujan identities, J. London Math. Soc., 4 (1929), 4–9.
G.N. Watson, A note on Lerch’s functions, Quart. J. Math., Oxford Ser., 8 (1937), 43–47.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1977 D. Reidel Publishing Company, Dordrecht-Holland
About this paper
Cite this paper
Andrews, G.E., Askey, R. (1977). Enumeration of Partitions: The Role of Eulerian Series and q-Orthogonal Polynomials. In: Aigner, M. (eds) Higher Combinatorics. NATO Advanced Study Institutes Series, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1220-1_1
Download citation
DOI: https://doi.org/10.1007/978-94-010-1220-1_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-1222-5
Online ISBN: 978-94-010-1220-1
eBook Packages: Springer Book Archive