Operator methods and Lagrange inversion: a unified approach to Lagrange formulas

Krattenthaler, Ch.

doi:10.1090/S0002-9947-1988-0924765-4

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Operator methods and Lagrange inversion: a unified approach to Lagrange formulas

Author: Ch. Krattenthaler
Journal: Trans. Amer. Math. Soc. 305 (1988), 431-465
MSC: Primary 05A30; Secondary 05A17, 11P57
MathSciNet review: 924765
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a general method of proving Lagrange inversion formulas and give new proofs of the -variable Lagrange-Good formula [13] and the -Lagrange formulas of Garsia [7], Gessel [10], Gessel and Stanton [11, 12] and the author [18]. We also give some -analogues of the Lagrange formula in several variables.

[1] Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545 (85g:33008)
[2] L. Carlitz, Some inverse relations, Duke Math. J. 40 (1973), 893–901. MR 0337651 (49 #2420)
[3] J. Cigler, Operatormethoden für 𝑞-Identitäten, Monatsh. Math. 88 (1979), no. 2, 87–105 (German, with English summary). MR 551934 (81h:05009), http://dx.doi.org/10.1007/BF01319097
[4] G. P. Egorychev, Integral representation and the computation of combinatorial sums, Translations of Mathematical Monographs, vol. 59, American Mathematical Society, Providence, RI, 1984. Translated from the Russian by H. H. McFadden; Translation edited by Lev J. Leifman. MR 736151 (85a:05008)
[5] L. Euler, De serie Lambertina plurimisque eius insignibus proprietatibus, Acta Acad. Sci. Petro, 1779; II, 1783, pp. 29-51; reprinted in Opera omnia, Ser. I, vol. 6, Teubner, Leipzig, 1921, pp. 350-369.
[6] J. Fürlinger and J. Hofbauer, 𝑞-Catalan numbers, J. Combin. Theory Ser. A 40 (1985), no. 2, 248–264. MR 814413 (87e:05017), http://dx.doi.org/10.1016/0097-3165(85)90089-5
[7] Adriano M. Garsia, A 𝑞-analogue of the Lagrange inversion formula, Houston J. Math. 7 (1981), no. 2, 205–237. MR 638947 (82m:05010)
[8] A. M. Garsia and S. A. Joni, A new expression for umbral operators and power series inversion, Proc. Amer. Math. Soc. 64 (1977), no. 1, 179–185. MR 0444487 (56 #2838), http://dx.doi.org/10.1090/S0002-9939-1977-0444487-4
[9] A. M. Garsia and S. A. Joni, Higher dimensional polynomials of binomial type and formal power series inversion, Comm. Algebra 6 (1978), no. 12, 1187–1215. MR 0491219 (58 #10484)
[10] Ira Gessel, A noncommutative generalization and 𝑞-analog of the Lagrange inversion formula, Trans. Amer. Math. Soc. 257 (1980), no. 2, 455–482. MR 552269 (82g:05006), http://dx.doi.org/10.1090/S0002-9947-1980-0552269-2
[11] Ira Gessel and Dennis Stanton, Applications of 𝑞-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), no. 1, 173–201. MR 690047 (84f:33009), http://dx.doi.org/10.1090/S0002-9947-1983-0690047-7
[12] Ira Gessel and Dennis Stanton, Another family of 𝑞-Lagrange inversion formulas, Rocky Mountain J. Math. 16 (1986), no. 2, 373–384. MR 843058 (87i:33007), http://dx.doi.org/10.1216/RMJ-1986-16-2-373
[13] I. J. Good, Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc. 56 (1960), 367–380. MR 0123021 (23 #A352)
[14] H. W. Gould and L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891. MR 0337652 (49 #2421)
[15] Josef Hofbauer, A 𝑞-analog of the Lagrange expansion, Arch. Math. (Basel) 42 (1984), no. 6, 536–544. MR 756895 (85i:05025), http://dx.doi.org/10.1007/BF01194051
[16] -, Lagrange-inversion (Seminaire Lotharingien de Combinatoire), I.R.M.A. n $^\circ$ 191/S-05, Strasbourg, 1982.
[17] S. A. Joni, Lagrange inversion in higher dimensions and umbral operators, Linear and Multilinear Algebra 6 (1978/79), no. 2, 111–122. MR 0491220 (58 #10485)
[18] Christian Krattenthaler, A new 𝑞-Lagrange formula and some applications, Proc. Amer. Math. Soc. 90 (1984), no. 2, 338–344. MR 727262 (85g:05022), http://dx.doi.org/10.1090/S0002-9939-1984-0727262-6
[19] -, -Lagrangeformel und inverse Relationen, Ph.D. dissertation, Vienna, 1983.
[20] Percy A. MacMahon, Combinatory analysis, Two volumes (bound as one), Chelsea Publishing Co., New York, 1960. MR 0141605 (25 #5003)
[21] Percy Alexander MacMahon, Collected papers. Vol. I, MIT Press, Cambridge, Mass., 1978. Combinatorics; Mathematicians of Our Time; Edited and with a preface by George E. Andrews; With an introduction by Gian-Carlo Rota. MR 514405 (80k:01065)
[22] John Riordan, Combinatorial identities, John Wiley & Sons Inc., New York, 1968. MR 0231725 (38 #53)
[23] S. S. Abhyankar, Lectures in algebraic geometry, Notes by Chris Christensen, Purdue Univ., 1974.
[24] Andrea Brini, Higher-dimensional recursive matrices and diagonal delta sets of series, J. Combin. Theory Ser. A 36 (1984), no. 3, 315–331. MR 744080 (86d:05013), http://dx.doi.org/10.1016/0097-3165(84)90039-6
[25] A. M. Garsia and S. A. Joni, Higher dimensional polynomials of binomial type and formal power series inversion, Comm. Algebra 6 (1978), no. 12, 1187–1215. MR 0491219 (58 #10484)
[26] Ira M. Gessel, A combinatorial proof of the multivariable Lagrange inversion formula, J. Combin. Theory Ser. A 45 (1987), no. 2, 178–195. MR 894817 (88h:05011), http://dx.doi.org/10.1016/0097-3165(87)90013-6
[27] P. Henrici, Die Lagrange-Bürmannsche Formel bei formalen Potenzreihen, Jahresber. Deutsch. Math.-Verein. 86 (1984), no. 4, 115–134 (German). MR 766156 (86d:30007)
[28] O. Viskov, Inversion of power series and the Lagrange formula, Soviet Math. Dokl. 22 (1980), 330-332.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|