An explanation of Mellin's 1921 paper

Wayne M. Lawton

The Bulletin of Irkutsk State University. Series Mathematics, Tome 46 (2023), pp. 98-109

Résumé

In 1921 Mellin published a Comptes Rendu paper computing the principal solution of a polynomial using generalized hypergeometric functions of its coefficients. He used an integral transform nowadays bearing his name. Slightly over three pages, the paper is written in French in a terse style befitting the language. This article makes Mellin's landmark result accessible to people who are not experts in hypergeometric functions and complex analysis by deriving detailed proofs that were omitted in Mellin's paper.

Détail

Keywords: functions of hypergeometric type, Mellin–Barnes integral representation.
Mots-clés : polynomial, principal solution

Bibliographie

[1] Aizenberg L.A., Yuzhakov A.P., Integral representations and residues in multidimensional complex analysis, Translations of Mathematical Monographs, 58, American Mathematical Society, Providence, RI, 1983 | DOI | MR | Zbl

[2] Aizenberg L.A., Tsikh A.K. and Yuzhakov A.P., “Multidimensional residues and applications”, Modern Problems of Mathematics, 8 (1985), 45–64

[3] Antipova I.A., “Inversion of many-dimensional Mellin transforms and solutions of algebraic equations”, Mat. Sbornik, 198:4 (2007), 447–463 | DOI | MR | Zbl

[4] Antipova L.A., Mikhalkin E.N., “Analytic continuation of a general algebraic function by means of Puiseux series”, Proc. Steklov Institute Math., 279 (2012), 3–13 | DOI | MR | Zbl

[5] Barnes E.W., “The asymptotic expansion of integral functions defined by generalized hypergeometric functions”, Proc. London Math. Soc., 5:2 (1907), 59–116 | DOI | MR

[6] Barnes E.W., “A new development of the theory of the hypergeometric functions”, Proc. London Math. Soc., 6:2 (1907), 141–177 | MR

[7] Barnes E.W., “On functions defined by simple types of hypergeometric series”, Trans. Camb. Phil. Soc., 20 (1907), 235–279

[8] Barnes E.W., “A transformation of generalised hypergeometric series”, Quart. J. Math., 141 (1910), 136–140

[9] Belardinelli G., Fonctions hypergéométriques de plusieurs variables et résolution analytique des équations algébriques générales, Mémorial des sciences mathématiques, 145, Gauthier–Villars, Paris, 1960 | MR

[10] Birkeland M.R., Ein allgemeiner Satz über algebraische Gleichungen, Annales Academiae Scientarum Fennicae. Ser. A, 7, 1915

[11] Debnath L., Bhatta D., Integral Transforms and Their Applications, 2nd ed., Chapman Hall/CRC, New York, 2007 | MR

[12] Euler L., Introduction to Analysis Infinitorum, v. 1, Lausanne, 1748 | MR

[13] Fosberg M., Passare M., Tsikh A., “Laurent determinants and arrangements of hyperplane amoebas”, Advances in Mathematics, 151 (2000), 45–70 | DOI | MR

[14] Grothendieck A., “Théorèms de dualité pour les faisceaux algébriques cohérent”, Séminaire Bourbaki, 49, 1957, 1–25 | MR

[15] Hartshorne R., Residues and duality, Lecture Notes in Math., 20, Springer-Verlag, Berlin–New York, 1966 | DOI | MR | Zbl

[16] Kauers M., Paule P., The Concrete Tetrahedron, Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, Springer, Vienna, 2011 | MR | Zbl

[17] Krantz S.G., Parks H.R., The Implicit Function Theorem. History, Theory, and Applications, Birkhäuser, Boston, 2003 | MR

[18] Kulikov V.R., Stepanenko V.A., “On solutions and Waring's formulas for systems of n algebraic equations for n unknowns”, St. Petersburg Math. J., 26:5 (2005), 839–848 | DOI | MR

[19] Mainardi F., Pagnini G., “Salvatore Pincherle: The Pioneer of the Mellin-Barnes Integrals”, J. Computational and Applied Mathematics, 153:1-2 (2003), 331–342 | DOI | MR | Zbl

[20] Mellin H.J., “Zur Theorie der Gamma Funktion”, Acta Math., 8 (1886), 37–80 | DOI | MR

[21] Mellin H.J., “Über einen Zusammenmenhang zwischen gewissen linearen Differential und Differenzengleichungen”, Acta Math., 9 (1886), 137–166 | DOI | MR

[22] Mellin H.J., “Zur Theorie der lineren Differenzengleichungen erster Ordnung”, Acta Math., 15 (1891), 317–384 | DOI | MR

[23] Mellin H.J., “Zur Theorie zweier allgemeiner Klassen bestimmter Integrale”, Acta Soc. Sc. Fenn., 22:2 (1896)

[24] Mellin H.J., “Résolution de l'equation algébrique générale á l'aide de la fonction gamma”, C. R. Acad. Sci. Paris Sér. I Math., 172 (1921), 658–661

[25] Mikhailkin E.N., “On solving general algebraic equations by integral of elementary functions”, Siberian Math. J., 47:2 (2006), 365–371 | DOI | MR

[26] Nilsson L., Passare M., Tsikh A.K., “Domains of convergence for A–hypergeometric series and integrals”, J. Siberian Federal Univ. Math. Physics, 12:4 (2019), 509–529 | DOI | MR

[27] Passare M., Tsikh A.K., Cheshel A.A., “Multiple Mellin-Barnes integrals as periods of Calabi-Yau manifolds with several moduli”, Theor. Math. Phys., 109 (1996), 1544–1555 | DOI | MR | Zbl

[28] Pincherle A., “Sopra una trasformazione delle equazioni differenziali lineari in equazioni lineari alle differenze, e viceversa”, R. Istituto Lombardo di Scienze e Lettere, Rendiconti, Ser. 2, 19 (1886), 559–562

[29] Pincherle A., “Sulle funzioni ipergeometriche generalizzate”, Atti R. Accademia Lincei, Rend. Cl. Sci. Fis. Mat. Nat., Ser. 4, 4 (1888), 694–700; 792–799

[30] Pincherle A., “Notices sur les travaux”, Acta Mathematica, 46 (1925), 341–362 | DOI | MR

[31] Sadykov T.M., Tsikh A.K., Hypergeometric and Algebraic Functions of Several Variables, Nauka Publ, M., 2014 (in Russian)

[32] Semusheva A.Y., Tsikh A.K., “Continuation of Mellin's Studies on Solving Algebraic Equations”, Complex Analysis and Differential Operators, On the Occasion of the 150th Anniversary of S. V. Kovalevskaya, Krasnoyarsk State University, 2000, 134 (in Russian) | MR

[33] Shabat B.V., Functions of Several Variables, v. II, Introduction to Complex Analysis, Am. Math. Soc., Providence, 1992 | MR | Zbl

[34] Slater L.J., Generalized Hypergeometric Functions, Cambridge University Press, UK, 1960 | MR

[35] Stepanenko V.A., “On the solution of a system of n algebraic equations with n unknowns by means of hypergeometric functions”, Vestnik Krasn. Gos. Univ. Ser. Fiz. Mat., 1 (2003), 35–48 (in Russian)

[36] Tsikh A.K., Multidimensional Residues and Their Applications, Am. Math. Soc., Providence, RI, 1992 | MR | Zbl

[37] Wallis J., Arithmeticum Infinitorum, London, 1655

[38] Yuzhakov A.P., “The application of the multiple logarithmic residue to the expansion of implicit functions in power series”, Mat. Sbornik, 97:2 (1975), 177–192 (in Russian) | DOI | MR | Zbl

[39] Zhdanov O.N., Tsikh A.K., “Studying the multiple Mellin–Barnes integrals by means of multidimensional residues”, Sib. Math. J., 39 (1998), 281–298 | DOI | MR | Zbl