Springer ramanujans notebooks part 3 (1991)

Ramanujan’s Notebooks Part III Bruce C Berndt Ramanujan’s Notebooks Part III Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Bruce C Berndt Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801 USA An earlier and shorter version of Chapter 16 was published in “Chapter 16 of Ramanujan’s second notebook: Theta-functions and q-series,” by C Adiga, B C Berndt, S Bhargava and G N Watson, Memoirs of the American Mathematical Society, Volume 53, Number 315, (January 1985) The revised version in this book appears by permission of the American Mathematical Society AMS Subject Classifications: 10-00, lo-03,01A6O,OlA75, lOAxx, 33-xx Library ofCongress Cataloging-in-Publication Data (Revised for Part 3) Ramanujan Aiyangar, Srinivasa, 1887-1920 Ramanujan’s notebooks Includes bibliographical references and indexes Mathematics I Berndt, Bruce C., 193911 Title QA3.R33 1985 510 84-20201 ISBN 0-387-96110-O (v 1) Printed on acid-free paper 1991 Springer-Verlag New York Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Typeset by Asco Trade Typesetting Ltd., Hong Kong Printed and bound by Edwards Brothers, Inc., Ann Arbor, MI Printed in the United States of America 987654321 ISBN o-387-97503-9 Springer-Verlag ISBN 3-540-97503-9 Springer-Verlag New York Berlin Heidelberg Berlin Heidelberg New York Dedicated to S Janaki Ammal (Mrs Ramanujan) S Janaki Ammal Photograph by B Berndt, 1987 A significant portion of G N Watson’s research was influenced by Ramanujan No less than thirty of Watson’s published papers were motivated by assertions made by Ramanujan in his letters to G H Hardy and in his notebooks Beginning in about 1928, Watson invested at least ten years to the editing of Ramanujan’s notebooks He never completed the task, but fortunately his efforts have been preserved Through the suggestion of R A Rankin and the generosity of Mrs Watson, all material pertaining to the notebooks compiled by Watson was donated to the library of Trinity College, Cambridge These notes were invaluable to the author in the preparation of this book In particular, many proofs in Chapters 19-21 are due to Watson We are grateful to the Master and Fellows of Trinity College, Cambridge for providing us a copy of Watson’s notes For an engaging biography of Watson, seeRankin’s paper [l] G N Watson Reprinted with courtesy of the London Mathematical Society 496 References Werke, Bd 3, Koniglichen Gesell Wiss., GBttingen, 1876, pp 413-432 Hundert Theoreme iiber die neuen Transscendenten, in Werke, Bd 3, Kiiniglichen Gesell Wiss., Gottingen, 1876, pp 461-469 Ghosh, S [l] On the Evaluation of k2 and E in the Theory of Elliptic Functions, Doctoral Dissertation, University of Bombay, 1985 Glaisher, J W L [l] Proof of the addition equation for elliptic integrals of the second kind by means of the q-series, Mess Math 12(1883), 43-48 [2] On the process of squaring the q-series for kp sn u, kp cn u, p dn u, Mess Math 16(1887), 145-149 [3] On the q-series derived from the elliptic and zeta functions of SK and *K, Proc London Math Sot 22(1891), 143-171 Glasser, M L [l] Definite integrals of the complete elliptic integral K, J Res Nat Bur Stand (B) Math Sci 80B (1976), 313-323 Glasser, M L., Privman, V., and Svrakic, N M [I] Temperley’s triangular lattice compact cluster model: exact solution in terms of the q series, J Phys Math Gen 20(1987), L12755L1280 Gordon, B [l] Some identities in combinatorial analysis, Q J Math (Oxford) (2) 12(1961), [2] [3] 285-290 Some continued fractions of the Rogers-Ramanujan type, Duke Math J 32(1965), 741-748 Gradshteyn, I S and Ryzhik, I M [l] Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965 Gray, J J [l] A commentary on Gauss’s mathematical diary, 1796-1814, with an English translation, Expos Math 2( 1984), 97- 130 Greenhill, A G [l] 7JreApplications of Elliptic Functions, Macmillan, London, 1892 [2] The seventeen-section of the elliptic function, Math Ann 68(1910), 208-219 Guetzlaff, C [ 1] Aequatio modularis pro transformatione functionurn ellipticarum septimi ordinis, J Reine Angew Math 12(1834), 173-177 Gustafson, R [l] The Macdonald identities for afhne root systems of classical type and hypergeometric series very-well poised on semisimple Lie algebras, in Ramanujan Znternational Symposium on Analysis, N K Thakare, ed., Macmillan India, Madras, 1989, pp 185-224 Hahn, W [l] Beitrlige zur Theorie der Heineschen Reihen, Math Nachr 2(1949), 340[2] 379 Hall, H S and Knight, S R [l] Higher Algebra, Macmillan, London, 1957 Halphen, M [ 1] Sur une formule ricurrents concernant les sommes des diviseurs des nombres entiers, Bull Sot Math France 5(1877), 158-160 Hancock, H [l] Elliptic Integrals, Wiley, New York, 1917 References 497 Hanna, M Cl] The modular equations, Proc London Math Sot (2) 28(1928), 46-52 Hansen, E R [l] A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975 Hardy, G H [I] Proof of a formula of Mr Ramanujan, Mess Math 44(1915), 18-21 [Z] Collected Papers, vol 5, Clarendon Press, Oxford, 1972 [3] Ramanujan, 3rd ed., Chelsea, New York, 1978 Hardy, G H and Wright, E M [l] An Introduction to the Theory of Numbers, 4th ed., Clarendon Press, Oxford, 1960 Hecke, E [I] 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summations due to Ramanujan, SIAM J Math Anal 9( 1978), 34-48 Littlewood, D E [1] A University Algebra, William Heinemann LTD, Melbourne, 1950 Macdonald, I [1] Affine root systems and Dedekind’s n-function, Znoent Math 15(1972), 91-143 Maclaurin, C [l] A Treatise ofFluxions in Two Books, vol 2, T W and T Ruddimans, Edinburgh, 1742 MacMahon, P A [1] Combinatory Analysis, vol 2, University Press, Cambridge, 1915 Mermin, N D [1] Pi in the sky, Am J Phys 55( 1987), 584-585 Milne, S C [1] An elementary proof of the Macdonald identities for A$‘), Adu Math 57( 1985), 34-70 [2] A U(n) generalization of Ramanujan’s i$i summation, Z Math Anal Appl 1180986) 263-277 [3] Multiple&series and U(n) generalizations of Ramanujan’s 1$1 sum, in Ramanujan Revisited, Academic Press, Boston, 1988, pp 473-524 [4] The multidimensional i$i sum and Macdonald identities for A$‘), in Theta Functions Bowdoin 1987, Part 2, Proceedings of Symposia in Pure Mathematics, vol 49, American Mathematical Society, Providence, 1989, pp 323-359 [S] Classical partition functions and the U(n + 1) Rogers-Selberg identity, Discrete Math., to appear Mimachi, K [1] A proof of Ramanujan’s identity by use of loop integral, SZAM J Math Anal 19(1988), 1490-1493 Moak, D S [1] The q-analogue of the Laguerre polynomials, J Math Anal Appl 81(1981), 20-47 500 [2] References The q-analogue of Stirling’s formula, Rocky Mountain Z Math 14(1984), 403-413 Mordell, L J [l] Note on certain modular relations considered by Messrs Ramanujan, Darling and Rogers, Proc London Math Sot (2) 20(1922), 408-416 [2] An identity in combinatorial analysis, Proc Glasgow Math Assoc 5(1962), 197-200 Moreau, C [l] Plus petit nombre &gal $ la somme de deux cubes de deux facons, LIZntermgdiaire Math 5( 1898), 66 Muir, T [l] On the perimeter of an ellipse, Mess Math 12(1883), 149-151 Nyvoll, M [l] Tilnaermelsesformler for ellipsebuer, Nordisk Mat Tidskr 25-26( 1978), 70-72 Odlyzko, A M and Wilf, H S [l] The editor’s corner: n coins in a fountain, Amer Math Monthly 95(1988), 840-843 Paule, P [l] On identities of the Rogers-Ramanujan type, J Math Anal Appl 107(1985), 255-284 Peano, G [l] Sur une formule d’approximation pour la rectification de l’ellipse, C R Acad Sci Paris 108(1889), 960-961 Perron, Band 2, dritte A&age, B G Teubner, [l] Die Lehre von den Kettenbriichen, Stuttgart, 1957 Petersson, H [l] Automorphic Forms and Discrete Groups, Unpublished lecture notes, University of Wisconsin, Madison, 1974 Preece, C T [l] Theorems stated by Ramanujan (VI): theorems on continued fractions, J London Math Sot 4(1929), 34-39 [2] Theorems stated by Ramanujan (XIII), Z London Math Sot 6(1931), 95-99 Privman, V and SvrakiC, N M [l] Difference equations in statistical mechanics I Cluster statistics models, J Stat Phys 51(1988), 1091-1110 Rademacher, H [l] Topics in Analytic Number Theory, Springer-Verlag, New York, 1973 Raghavan, S [l] On certain identities due to Ramanujan, Q J Math (Oxford) (2) 37(1986), 221-229 [2] Euler products, modular identities and elliptic integrals in Ramanujan’s manuscripts, in Ramanujan Revisited, Academic Press, Boston, 1988, pp 335-345 Raghavan, S and Rangachari, S S [l] On Ramanujan’s elliptic integrals and modular identities, in Number Theory and Related Topics, Oxford University Press, Bombay, 1989, pp 119-149 Rahman, M [l] A simple evaluation of Askey and Wilson’s q-beta integral, Proc Amer Math Sot 92(1984), 413-417 Ramamani, V [l] Some Identities Conjectured by Srinivasa Ramanujan Found in His Lithographed References 501 Notes Connected with Partition Theory and Elliptic Modular Functions-Their Proofs-Inter Connection with Various Other Topics in the Theory of Numbers and Some Generalizations, Doctoral Thesis, University of Mysore, 1970 Ramamani, V and Venkatachaliengar, K [l] On a partition theorem of Sylvester, Michigan Math J 19(1972), 137-140 Ramanathan, K G [ 1] On Ramanujan’s continued fraction, Acta Arith 43( 1984), 209-226 [2] On the Rogers-Ramanujan continued fraction, Proc Indian Acad Sci (Math Sci.) 93(1984), 67-77 [3] Remarks on some series considered by Ramanujan, J Indian Math Sot 46(1982), 107-136 (published in 1985) [4] Ramanujan’s continued fraction, Indian J Pure Appl Math 16(1985), 695-724 [S] Some applications of Kronecker’s limit formula, J Indian Math Sot 52(1987), 71-89 [6] Hypergeometric series and continued fractions, Proc Indian Acad Sci (Math Sci.) 97( 1987), 277-296 [7] Ramanujan’s notebooks, J Indian Inst Sci., Special Issue (1987), 25-32 [S] Generalisations of some theorems of Ramanujan, J Number Theory 29(1988), 118-137 [9] On some theorems stated by Ramanujan, in Number Theory and Related Topics, Oxford University Press, Bombay, 1989, pp 151- 160 [lo] Ramanujan’s modular equations, Acta Arith 53( 1990), 403-420 Ramanujan, S [l] Squaring the circle, J Indian Math Sot 5( 1913), 132 [2] Modular equations and approximations to 7t, Q J Math (Oxford) 45(1914), 350-372 [3] Question 584, J Indian Math Sot 6(1914), 199 [4] Some definite integrals, Mess Math 44(1915), 75-85 [S] Question 662, I Indian Math Sot 7(1915), 119-120 [6] On certain arithmetical functions, Trans Cambridge Philos Sot 22(1916), 159-184 [7] Question 755, J Indian Math Sot 8(1916), 80 [8] Proof of certain identities in combinatory analysis, Proc Cambridge Philos Sot 19(1919), 214-216 [9] Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957 [lo] Collected Papers, Chelsea, New York, 1962 [l l] ‘Ihe Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988 Ramanujan Centenary Prize Competition [l] Mathematics Today 5(1987), Nos 2-3,21-24; No 4,45-48; No 5,42-46, Rankin, R A [l] George Neville Watson, J London Math Sot 41(1966), 551-565 [2] Modular Forms and Functions, Cambridge University Press, Cambridge, 1977 Rao, M B and Ayyar, M V [l] On some infinite products and series II, J Indian Math Sot 15(1923-24X 233-247 Riesel, H [l] Some series related to infinite series given by Ramanujan, BZT13(1973), 97-l 13 Rogers, L J [l] Second memoir on the expansion of certain infinite products, Proc London Math Sot 25(1894), 318-343 [2] Third memoir on the expansion of certain infinite products, Proc London Math Sot 26(1895), 15-32 References 502 [3] On the representation of certain asymptotic series as convergent continued fractions, Proc London Math Sot (2) 4(1907), 72-89 [4] On two theorems of combinatory analysis and some allied identities, Proc London Math Sot (2) 16(1917), 315-336 [S] Proof of certain identities in combinatory analysis, Proc Cambridge Philos Sot 19(1919), 211-214 [6] On a type of modular relation, Proc London Math Sot (2) 19(1920), 387-397 Rothe, H A Systematisches Lehrbuch der Arithmetik, Barth, Leipzig, 1811 [l] Russell, R [l] On rc1- rc’r2’modular equations, Proc London Math Sot 19( 1887), 90-l 11 [2] On modular equations, Proc London Math Sot 21(1890), 351-395 Schllfli, L [l] Beweis der Hermiteschen Verwandlungstafeln fiir die elliptischen Modularfunctionen, Z Reine Angew Math 72(1870), 360-369 Schoeneberg, B [ 1] Elliptic Modular Functions, Springer-Verlag, New York, 1974 Schoissengeier, J [l] Der numerische Wert gewisser Reihen, Manuscripta Math 38(1982), 257-263 Schroter, H [l] De aequationibus modularibus, Dissertatio Inauguralis, Albertina Litterarum Universitate, 1854, Regiomonti [2] Extrait dune lettre adressee A M Liouville, J Math (2) 3(1858), 258-264 [3] Ueber Modulargleichungen der elliptischen Functionen, Auszug aus einem Schreiben an Herrn L Kronecker, J Reine Angew Math 58(1861), 378-379 [4] Beitrage zur Theorie der elliptischen Funktionen, Acta Math 5(1894), 205-208 Schur, I J [l] Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbriiche, S.-Z? Preuss Akad Wiss Phys.-Math Kl (1917), 302-321 [Z] Zur additiven Zahlentheorie, S.-Z? Preuss Akad Wiss Phys.-Math Kl (1926), [3] 488-495 Gesammelte Abhandlungen (3 volumes), Springer-Verlag, Berlin, 1973 Schwarz, H A [l] Formeln und Lehrsiitze zum Gebrauche der Elliptischen Funktionen Nach Vorlesungen und Aufzeichnungen des Herrn Prof K Weierstrass, Zweite Ausgabe, Erste Abteilung, Springer, Berlin, 1893 Sears, D B [l] Two identities of Bailey, J London Math Sot 27( 1952), 510-511 Selberg, A [l] Uber einige arithmetische Identitlten, Auh Norske Vid.-Akad Oslo I Mat.Naturv Kl., No 8, (1936), 3-23 [2] Collected Papers, vol 1, Springer-Verlag, Berlin, 1989 Selmer, E S [l] Bemerkninger til en ellipse-beregning av en ellipses omkrets, Nordisk Mat Tidskr 23(1975), 55-58 Singh, S N [l] Basic hypergeometric series and continued fractions, Math Student 56(1988), 91-96 Slater, L J [l] Further identities of the Rogers-Ramanujan type, Proc London Math Sot (2) 54(1952), 147-167 References 503 Sohncke, L A [l] Aequationes modulares pro transformatione functionum ellipticarum et undecimi et decimi tertii et decimi septimi ordinis, J Reine Angew Math 12(1834), 178 [2] Aequationes modulares pro transformatione functionum ellipticarum, J Reine Angew Math 16(1837), 97-130 Srivastava, H M [l] A note on a generalization of a q-series transformation of Ramanujan, Proc Japan Acad Ser A 63(1987), 143-145 Stanton, D [l] An elementary approach to the Macdonald identities, in q-Series and Partitions, D Stanton, ed., IMAvolume in Mathematics and Its Applications, vol 18, Springer-Verlag, New York, 1989, pp 139-149 Stark, H M [l] L-functions at s = IV First derivatives at s = 0, Ado Math 35(1980), 197235 Stieltjes, T J Cl] Sur la reduction en fraction continue dune serie procedant suivant les puissances descendantes dune variable, Ann Fat Sci Toulouse 3( 1889), l- 17 [2] Oeuures Complhes, t 2, P Noordhoff, Groningen, 1918 Stubban, J [ 11 Fergestads formel for tilnaermet beregning av en ellipses omkrets, Nordisk Mat Tidskr 23(1975), 51-54 Subbarao, M V and Vidyasagar, M [l] On Watson’s quintuple product identity, hoc Amer Math Sot 26(1970), 23-27 Szekeres,G [l] A combinatorial interpretation of Ramanujan’s continued fraction, Can Math Bull 1l( 1968), 405-408 Tannery, J and Molk, J [ 11 Eliments de la Theorie des Fonctions Elliptiques, t 2, Chelsea, New York, 1972 Thiruvenkatachar, V R and Venkatachaliengar, K [l] Ramanuian at Elementary Levels; Glimpses, to appear Venkatachaliengar, K [l] Development of Elliptic Functions According to Ramanujan, Tech Rep 2, Madurai Kamaraj University, Madurai, 1988 Verma, A [l] On identities of Rogers-Ramanujan type, Indian J Pure Appl Math 1l( 1980), 770-790 [2] Basic hypergeometric Srinivasa Ramanujan series and identities of Rogers-Ramanujan type, in (1887-1920), Macmillan India Ltd., Madras, 1988, pp 66-74 Verma, A., Denis, R Y., and Rao, K S [l] New continued fractions involving basic hypergeometric sQ2functions, J Math Phys Sci 21(1987), 585-592 Verma, A and Jain, V K [l] Transformations between basic hypergeometric series on different bases and identities of Rogers-Ramanujan type, J Math Anal Appl 76(1980), 230-269 [2] Transformations of non-terminating basic hypergeometric series, their contour integrals and applications to Rogers-Ramanujan identities, Z Math Anal Appl 87(1982), 9-44 504 References Viete, F [l] Opera Mathematics, Georg Olms Verlag, Hildesheim, 1970 Villarino, M B [l] Ramanujan’s approximation to the arc length of an ellipse, preprint Wall, H S [l] Analytic Theory ofContinued Fractions, van Nostrand, New York, 1948 Watson, G N Cl1 Theorems stated by Ramanujan (II): theorems on summation of series, Z London Math Sot 3(1928), 216-225 PI A new proof of the Rogers-Ramanujan identities, J London Math Sot 4(1929), 4-9 c31 Theorems stated by Ramanujan (VII): theorems on continued fractions, Z London Math Sot 4(1929), 39-48 c41 Theorems stated by Ramanujan (IX): two continued fractions, J London Math Sot 4(1929), 231-237 PI Ramanujan’s note books, J London Math Sot 6(1931), 137-153 C61 Ramanujan’s continued fraction, Proc Cambridge Philos Sot 31( 1935), 7- 17 Weber, H [l] Zur Theorie der elliptischen Functionen, Acta Math 11(1887-88), 333-390 [2] Elliptische Functionen und Algebraische Zahlen, Friedrich Vieweg und Sohn, Braunschweig, 1891 [3] Lehrbuch der Algebra, dritter Band, Friedrich Vieweg und Sohn, Braunschweig, 1908 Whittaker, E T and Watson, G N [ 11 A Course of Modern Analysis, ed., University Press, Cambridge, 1966 Wolfram, J [l] Tables of Natural Logarithms, included in G Vega, Ten Place Logarithms, reprint of 1794 edition, Hafner, New York, 1958 Woyciechowsky, J [l] Sipos Pal egy kezirata es a kochleoid, Mat Fiz Lapok 41(1934), 45-54 Zhang, L.-C [l] q-Difference equations and Ramanujan-Selberg continued fractions, Acta Arith 57( 1991), to appear Zucker, I J [l] The summation of series of hyperbolic functions, SIAM J Math Anal 10(1979), 192-206 [2] Some infinite series of exponential and hyperbolic functions, SIAM J Math Anal 15(1984), 406-413 [3] Further relations amongst infinite series and products II The evaluation of 3-dimensional lattice sums, J Phys A: Math Gen 23(1990), 117-132 Index Adiga, C 7, 10, 18, 24, 28, 32, 39, 49, 73, 79, 86, 115, 142 Agarwal, A K 78 Alder, H L 78 Almkvist, G 10,147,149-150,456 Al-Salam, W A 79 Andrews, G E 10, 13-15,18,28-29, 32,36-37,77-80,83,347,398 Askey, R 2, 10, 14,29,32,77-79, 149 Atkin, A L 83 Ayyar, M V 140 Bailey, D H Bailey, W N 15-17, 83, 89, 111, 120, 181,262 base (of an elliptic function) 5, 102 basic hypergeometric series 12 Baxter, R J 78 Berndt, B C 2, 5,7, 24,29,44, 79, ill-112,140-141,147,150,172, 197,326,456 Bernoulli numbers 42,61-64,97 Berry, A 440 Bhagirathi, N A 28 Bhargava, S 7, 10, 18, 24,28, 32, 39, 49, 73, 79, 115, 142 Biagioli, A J 7, 10, 326, 346 Biecksmith, R 73,83 Borwein, J M 6,78,269, 305, 346, 355,456 Borwein, P B 6,78,269, 305, 346, 355, 456 Bressoud, D 78, 398 Brillhart, J 10, 73, 83 Brouncker, Lord 200 Burnside, W 240 Byrd, P F 113 Carlitz, L 18, 79, 83 Catalan’s constant 154-155 Cauchy, A 14, 140 Cayley, A 2-3, 5, 106-107, 135, 138, 218,220,232,241 change of sign 126,130,132,178 Chudnovsky, D V 168 Chudnovsky, G V 168 Churchhouse, R F 79 Clausen transformation 114 Cohn, H column-row method of summation 114 complementary modulus 4, 102 complete series 42,455 consistency condition 327 continued fraction, geometric and arithmetic mean arguments 164 506 continued fractions 19-29,92,146, 151,163-168,185187,206-208, 221-222,345-347 cosine identity 345347-348 Court, N A 245 Coxeter, H S M 245 cusp 328 cusp parameter 328 Darling, H B C 261 Dedekind eta-function 37,44, 330338 degree of a modular equation degree of a modulus 229 degree of series 42 Denis, R Y 28,78-79 Deutsch, J Dickson, L E 197,200 Digby, K 200 dimidiation 126, 178 diophantine equations 197-200 divisor functions 62,6&65 duplication 125, 127-128, 178 Dyson, F J lo,83 eccentricity of an ellipse 145 Ehrenpreis, L 78 Eisenstein, G 28 Eisenstein series 7, 65, 121-122, 126-139,144-145,175-177,454488 Eisenstein series, values in terms of elliptic function parameters 126129 ellipse, approximations to the perimeter of 145-150,180-189 elliptic curve elliptic functions 2-3 elliptic functions, notation 101-102 elliptic integral of the first kind 4, 102 elliptic integral of the second kind 176-177,303-304 elliptic integrals 104 113, 238-243, 297-298 elliptic integrals, addition theorem for elliptic integrals of the first kind 106-108 Index elliptic integrals, addition theorem for elliptic integrals of the second kind 303 elliptic integrals, duplication formula 106 Enneper, A 5,72,220 Euler, L 14,37, 147, 150, 196-197, 199 Euler numbers 61,63 Euler’s diophantine equation 197-199 Euler’s partition theorem 37 Euler’s pentagonal number theorem 36-37 Evans, R J 7, 10, 83, 274, 276, 337, 352,373,375 F(x) 91 Fergestad, J B 146 Fermat, P 200 Fiedler, E 5, 315, 364,416,444 Fine, N J 32 fixed point of a modular form 328 Flajolet, P 80, 168 Forrester, P J 78 Forsyth, A R 242 Francon, J 168 Frenicle 200 Fricke involution 216,404 Fricke, R 5,83,364,416 Friedman, M D 113 Frobenius, G fundamental set 328 Garsia, A M 78 Gauss, C F 14,28, 36, 89, 147, 151, 181 Gauss’ transformation 113 geometrical problems 190-196,211213,243-249,298-302 Gerst, I 73, 83 Ghosh, S 488 Glaisher, J W L 169,242, 303 Glasser, M L 80, 110, 113 Gordon, B 78-79,83,347 Gosper, R W 13 Gray, J J 28 Greenhill, A G 2,400,439 Index Guetzlaff, C 5, 315 Gustafson, R 32 Hahn, W 32 Halphen, M 62 Hancock, H 212 Hanna, M $440,444 hard hexagon model 78 Hardy, G H 2, 6, 9, 11, 29-32, 36, 39, 45,77,79,84,86,126,162-B%, 197, 199,262,326,346,385,426,450 Hecke, E 398 Hecke operator 373 Heine, E 11, 14-15, 18,21 Heine’s continued fraction 21 Hermite, C 135 Hirschhorn, M 11,28, 31, 79,83, 347 Hoppe, R 196 Hovstad, R M 79 Hurwitz, A 5,444 hyperbola, perimeter of 180 hyperbolic function series evaluations in closed form 140-141,157-162 hyperbolic function series evaluations in terms of elliptic function parameters 132-139, 153-157, 172-178 hyperbolic function series identity 162 hypergeometric differential equation 120-121 hypergeometric functions 3,5,88-104, 120-122,144-150,153-155,164, 185-186,188,213,238-239,289290,455-456 invariant order of a modular form inversion formula for base q 100 Ismail, M E H 32,42, 79 Ivory, J 146-147 328 Jackson, F H 14-15 Jackson, M 32 Jacobi, C G J 3, 5, 11, 14, 36, 39, 54, 87, 115-116,123,126, 135,143, 165-166,169,173,176-177,207, 218,220,232,234,239-241 507 Jacobi triple product identity 11-12, 32,35-36 Jacobian elliptic functions 3, 54,87, 107-108,135-136,138-139,143, 162-163,165-180,207-208,227, 242,304 Jacobian elliptic functions, conversions of old formulas into new formulas 173-174 Jacobi’s identity 39 Jacobi’s imaginary transformation 106, 154 Jacobsen, L 10,20,22-24,26-27,79, 84,146 Jain, V K 78 Journal of the Indian Mathematical Society 9, 11,77, 190,246 Joyce, G S Kac, V G 398 Kepler, J 147, 150 Kiper, A 169 Kleiber, J 89 Klein, F 5, 315, 377,444 Knopp, M I 42,44,327,330 Koblitz, N I 29 Kohler, G Kondo, T 6,72,366 Koomwinder, T H 13 Kumbakonam Lamphere, R 10,29,79 Landen, J 5,146,181 Landen’s transformation 113, 126, 146-147,213 Langebartel, R 169 lattice gases Legendre, A M 5,107,181,220,232, 234,244 Legendre functions 89 Legendre-Jacobi symbol 329 Legendre’s relation 455-456 Lepowsky, J 78 Lie algebras 78 Ling, C.-B 140, 142 Littlewood, D E 311 Littlewood, J E 508 Index Macdonald, I 32 Macdonald identities 32 Maclaurin, C 146 MacMahon, P A 113 MACSYMA 10,312,369,372,377, 400,408,416-417,425,430,488 Mathematics 10 medial section 298 Mehler-Dirichlet integral 89 Mermin, N D 151 Metius, A 194 Milne, S 32, 78 Mimachi, K 32 \Mittag-Leffler theorem 144 ,nixed modular equation, deiintion 325 mixed modular equations, table of degrees 325-326 Moak, D S 29 modular equation, definition 213 modular equations 3-8 modular equations of degree 214 modular equations of degree 230238,352-353,356 modular equations of degree 214215 modular equations of degree 280288 modular equations of degree 314324,435-437 modular equations of degree 216217 modular equations of degree 352358 modular equations of degree 11 363372 modular equations of degree 13 376377 modular equations of degree 15 383397,435-439 modular equations of degree 16 216 modular equations of degree 17 397400 modular equations of degree 19 416417 modular equations of degree 21 400408 modular equations of degree 23 41 l416 modular equations of degree 25 290297 modular equations of degree 27 360362 modular equations of degree 31 439444 modular equations of degree 33 40% 411 modular equations of degree 35 423426,430 modular equations of degree 39 426430,435-439 modular equations of degree 47 444449 modular equations of degree 55 426430,435-439 modular equations of degree 63 426435,435-439 modular equations of degree 71 444449 modular equations of degree 87 449453 modular equations of degree 95 430435 modular equations of degree 119 430435 modular equations of degree 135 430435,449-453 modular equations of degree 143 430435 modular equations of degree 175 449453 modular equations of degree 207 449453 modular equations of degree 23 449453 modular equations of degree 247 449453 modular equations of degree 255 449453 modular equations, table 8,325-326 modular form, definition 328 modular forms 7,326-345,366-376, 399-408,415-417,423-425,430, 484-488 modular group 327 modulus 4-5,102 Molk, J 6,45, 72 Index Mordell, L J 2, 83, 261 Moreau, C 200 Muir, T 147, 150 Miiller, R 10 multiplier 5,214, 230 multiplier system 7,328-329 multiplier system of Dedekind eta-function 330 multiplier systems for theta-functions 330-332 National Science Foundation 10 notation 10, 12, 88-89,230-231, 326329 Nyvoll, M 147 Odlyzko, A M 79 order of a modular form 328 orders of theta-functions at rational cusps 333 partial fraction expansions 200-206 partition function 262 Paule, P 78 Peano, G 147, 150 pendulum 212-213,243-244,246, 299-301 perfect series 42 Perron, 166-167,186,208 Peterson, D H 398 Petersson, H 326 Pfaff’s transformation 17 pi, approximations to 151- 152, 194196 Playfair, J 146 Preece, C T 163-164 Privman, V 80 Proceedings of the London Mathematical Society 77 psi function 88, 90 pure series 42 Purtilo, J M 7, 10, 326 q-analogue of Dougall’s theorem 15 509 q-analogue of Gauss’ theorem 14 q-beta integral 11,29 q-binomial theorem 14, 32 q-gamma function 13 q-series 11-12, 14-19,21-34 quintic algorithm for calculating pi 269 quintuple product identity 11, 32, 56-57,59,80-83,338 Rademacher, H 218,273,330 Raghavan, S 7, 113,262-263,324 Rahman, M 29 Rama Murthy, C 10 Ramamani, V 18,31,54 Ramanathan, K G 4,7, 10,20, 28,79, 82,84,86,221-222,262,265,274, 276,324,347 Ramanujan Centenary Prize Competition 246 Ramanujan’s r$i summation 11,3134 Ramanujan’s quarterly reports 29 Ramanujan’s theta-function 18 Rangachari, S S 7,113,263,324 Rankin, R A 326,328-329,332-333, 342,370,373,404,484 Rao, K S 79 Rao, M B 140 reciprocal of a modular equation 216 reciprocal relation 334 Riesel, H 140 Rogers, L J 14, 18, 30, 77-79, 144, 163,166-168,207,398 Rogers-Ramanujan continued fraction 11,30-31,79-80,267 Rogers-Ramanujan continued fraction, combinatorial interpretation 7980 Rogers-Ramanujan continued fraction, finite form 31 Rogers-Ramanujan identities 11, 7779 Rothe, H A 14 row-column method of summation 114 Roy, R 29 510 Index Russell, R 315, 364, 377,400,416,435, 439,444 Schllfli, L 5, 315, 364, 377,400,416 Schoeneberg, B 328,342,368,484 Schoissengeier, J 142 Schroter, H 5-7,66,72,315,364,411, 439-440 Schroter’s formulas 6-7, 11,66-74 Schur, I J 77 Schwarz, H A 83 Scoville, R 79 Sears, D B 83 Selberg, A 79,347 Selmer, E S 146-147, 150 septic algorithm for calculating pi 305 Singh, S N 79 Sipos, P 147,150 Slater, L J 78 Sohncke, L A 5,315,364,377,400,416 Somashekara, D D 39,73,79 squaring the circle 193-195 Srivastava, H M 18 Stanton, D 32 Stark, H M 337,339 Stieltjes, T J 144, 163, 166-168, 207 Stolarsky, K 10 stroke operator 326 Stubban, J 146 Subbarao, M V 83 summation by rows or columns 113114 SvrakiC, N M 80 Swinnerton-Dyer, P 83 Szekeres, G 79 Tannery, J 6,45,72 Tartaglia 354 Tasaka, T 6,72,366 Tata Institute 10 taxi cab story 199 theta-functions 3-7, 11-12, 34,98104,114-125,139-141,218-219, 221-238,249-297,302-324,330334,337-488 theta-functions, basic identities 39-41, 43-52 theta-functions, logarithms of 38 theta-functions, values 103-104,210 theta-functions, values in terms of elliptic function parameters 122125 theta-function transformation formulas 36,43-44,102,208-209 Thiruvenkatachar, V R 8,32,34, 104 triplication formula 238-241 University of Illinois University of Madras University of Mysore 10 86 valence formula 329, 334, 336 Vaughn Foundation 10 Venkatachaliengar, K 8, 18, 32, 34, 104 Verma, A 78-79 Vidyasagar, M 83 Vitte, F 197 Villarino, M 10, 150, 184, 190 Waadeland, H 1492,146 Wall, H S 28 Wallis, J 200 Watson, G N 6, 10-11, 16, 19, 24, 30, 77,83-84,162,194,198,244,346 Weber, H 5,385,416,425-426,440, 444 Weierstrass, K 83 weight of a modular form 328 Wetzel, J 10 width of a subgroup of the modular group at a cusp 328 Wilf, H S 79 Wilson, J 29 Wilson, R L 78 Wolfram, J 151 Woyciechowsky, J 147 Wright, E M 36, 39, 197 Zeilberger, D 78 Zhang, L.-C 347 Zucker, I J 6,10,140,142,262 ... 537 11, 13 18,19 21 21 12 16 15 22 23 23 15 11 13 14 19,21 20 24 18,19 19,21 20 24 19,21 20 24 20 24 20 24 24 24 11 13 15 17 19 23 31 47 71 39 9 5, 25 3, 5, 15 3, 7, 21 3, 9, 27 3, 11, 33 3, 13, ... 21 3, 9, 27 3, 11, 33 3, 13, 39 3, 21, 63 3, 29, 87 5, 7, 35 5, 11, 55 5, 19,95 5, 27, 135 7, 9, 63 7, 17, 119 7, 25, 175 9, 15, 135 9, 23, 207 11, 13, 1 43 l&21, 231 13, 19,247 15,17,255 Each of Chapters... xxviii, (6) p xxix, (15) p xxix, (20) (i), (v) p xxix, (21) P 35 0, (3) p 35 3, (20) (ii), (iii), (iv), (vi) p 35 3, (21) p 35 3, (22) Location in Notebooks Chapter Chapter Chapter Chapter Chapter Chapter