Applications

The Rogers–Ramanujan identities appeared in Baxter’s solution of thehard hexagon modelin statistical mechanics.

Ramanujan’s continued fractionis

1 + q

1 + ^q2

1+_1+···^q3

= G(q) H(q).

31.5 See also

• Rogers polynomials

31.6 References

• Rogers, L. J.;Ramanujan, Srinivasa(1919), “Proof of certain identities in combinatory analysis.”, Cambr. Phil.

Soc. Proc. 19: 211–216, Reprinted as Paper 26 in Ramanujan’s collected papers

• Rogers, L. J. (1892), “On the expansion of some infinite products”, Proc. London Math. Soc. 24 (1): 337–352, doi:10.1112/plms/s1-24.1.337,JFM 25.0432.01

• Rogers, L. J. (1893), “Second Memoir on the Expansion of certain Infinite Products”, Proc. London Math.

Soc. 25 (1): 318–343,doi:10.1112/plms/s1-25.1.318

• Rogers, L. J. (1894), “Third Memoir on the Expansion of certain Infinite Products”, Proc. London Math. Soc.

26 (1): 15–32,doi:10.1112/plms/s1-26.1.15

• Issai Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, (1917) Sitzungsberichte der Berliner Akademie, pp. 302–321.

• W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.

• George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Math-ematics and Its Applications, 96, Cambridge University Press, Cambridge.ISBN 0-521-83357-4.

• Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son, The Rogers-Ramanujan Continued Fraction, J. Comput. Appl. Math. 105 (1999), pp. 9–24.

• Cilanne Boulet, Igor Pak,A Combinatorial Proof of the Rogers-Ramanujan and Schur Identities, Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030.

• Slater, L. J. (1952), “Further identities of the Rogers-Ramanujan type”, Proceedings of the London Mathemat-ical Society. Second Series 54 (2): 147–167,doi:10.1112/plms/s2-54.2.147,ISSN 0024-6115,MR 0049225

112 CHAPTER 31. ROGERS–RAMANUJAN IDENTITIES

31.7 External links

• Weisstein, Eric W.,“Rogers-Ramanujan Identities”,MathWorld.

• Weisstein, Eric W.,“Rogers-Ramanujan Continued Fraction”,MathWorld.

Chapter 32

Schwarz’s list

In the mathematical theory ofspecial functions, Schwarz’s list or the Schwartz table is the list of 15 cases found byHermann Schwarz(1873, p. 323) whenhypergeometric functionscan be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which thehypergeometric equationhas a finitemonodromy group, or equivalently has two independent solutions that arealgebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of acyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certainspherical triangles.

The wider importance of the table, for general second-order differential equations in the complex plane, was shown by Felix Klein, who proved a result to the effect that cases of finite monodromy for such equations andregular singularities could be attributed to changes of variable (complex analytic mappings of theRiemann sphereto itself) that reduce the equation to hypergeometric form. In fact more is true: Schwarz’s list underlies all second-order equations with regular singularities on compactRiemann surfaceshaving finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic mapping, of degree computable from the equation’s data.^[1][2]

The numbers λ, μ, ν are (up to a sign) the differences 1 − c, c − a − b, a − b of the exponents of thehypergeometric differential equationat the three singular points 0, 1, ∞. They are rational numbers if and only if a, b and c are, a point that matters in arithmetic rather than geometric approaches to the theory.

32.1 Further work

An extension of Schwarz’s results was given by T. Kimura, who dealt with cases where theidentity componentof thedifferential Galois groupof the hypergeometric equation is asolvable group.^[3][4]A general result connecting the differential Galois group G and the monodromy group Γ states that G is theZariski closureof Γ — this theorem is attributed in the book of Matsuda toMichio Kuga. By general differential Galois theory, the resulting Kimura-Schwarz table classifies cases of integrability of the equation by algebraic functions andquadratures.

Anotherrelevant list is that of K. Takeuchi, who classified the (hyperbolic)triangle groupsthat arearithmetic groups (85 examples).^[5]

Émile Picardsought to extend the work of Schwarz in complex geometry, by means of ageneralized hypergeometric function, to construct cases of equations where the monodromy was adiscrete groupin theprojective unitary group PU(1, n).Pierre DeligneandGeorge Mostowused his ideas to constructlatticesin the projective unitary group. This work recovers in the classical case the finiteness of Takeuchi’s list, and by means of a characterisation of the lattices they construct that are arithmetic groups, provided new examples of non-arithmetic lattices in PU(1, n).^[6]

Baldassari applied the Klein universality, to discuss algebraic solutions of theLamé equationby means of the Schwarz list.^[7]

32.2 See also

• Schwarz triangle

113

114 CHAPTER 32. SCHWARZ’S LIST

32.3 Notes

[1] A modern treatment is in F. Baldassarri, B. Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math. 101 (1) (1979) 42–76.

[2] http://archive.numdam.org/ARCHIVE/GAU/GAU_1986-1987__14_/GAU_1986-1987__14__A12_0/GAU_1986-1987_

_14__A12_0.pdf, pp.5-6.

[3] http://fe.math.kobe-u.ac.jp/FE/Free/vol12/fe12-18.pdf

[4] http://www.intlpress.com/MAA/p/2001/8_1/MAA-8-1-113-120.pdfat p. 116 for the formulation.

[5] http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1240433796 [6] http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1986__63_/PMIHES_1986__63__5_0/PMIHES_1986__63_

_5_0.pdf

[7] F. Baldassarri, On algebraic solutions of Lamé’s differential equation, J. Differential Equations 41 (1) (1981) 44–58. Cor-rection inAlgebraic Solutions of the Lamé Equation, Revisited (PDF), by Robert S. Maier.

32.4 References

• Matsuda, Michihiko (1985), Lectures on algebraic solutions of hypergeometric differential equations(PDF), Lectures in Mathematics 15, Tokyo: Kinokuniya Company Ltd.,MR 1104881

• Schwarz, H. A. (1873), “Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt”,Journal für die reine und angewandte Mathematik75:

292–335,ISSN 0075-4102

32.5 External links

• Towards a nonlinear Schwarz’s list (PDF)

Chapter 33

Wilson polynomials

In mathematics, Wilson polynomials are a family oforthogonal polynomialsintroduced byJames A. Wilson(1980) that generalizeJacobi polynomials,Hahn polynomials, andCharlier polynomials.

They are defined in terms of thegeneralized hypergeometric functionand thePochhammer symbolsby

pn(t²) = (a + b)n(a + c)n(a + d)n4F3

( −n a + b + c + d + n− 1 a − t a + t

a + b a + c a + d ; 1

) .

33.1 See also

• Askey-Wilson polynomialsare a q-analogue of Wilson polynomials.

33.2 References

• Wilson, James A. (1980), “Some hypergeometric orthogonal polynomials”, SIAM Journal on Mathematical Analysis 11 (4): 690–701,doi:10.1137/0511064,ISSN 0036-1410,MR 579561

• Koornwinder, T.H. (2001), “Wilson polynomials”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

115

116 CHAPTER 33. WILSON POLYNOMIALS