On complex solutions of certain partial differential equations

References

• Berenstein CA, Li BQ. On certain first-order partial differential equations in Cn, Harmonic analysis, signal processing, and complexity. Boston: Birkhäuser; 2005. (Progress in mathematics, Vol. 238, p. 29–36).
   Google Scholar
• Chang DC, Li BQ. Description of entire solutions of eiconal type equations. Can Math Bull. 2012;55:249–259.
   Web of Science ®Google Scholar
• Han Q. On complex analytic solutions of the partial differential equation Cn. Houston J Math. 2009;35:277–289.
   Web of Science ®Google Scholar
• Hemmati J. Entire solutions of first-order partial differential equations. Proc Am Math Soc. 1997;125:1483–1485.
   Web of Science ®Google Scholar
• Hu PC, Yang CC. Global solutions of homogeneous linear partial differential equations of the second order. Michigan Math J. 2009;58:807–831.
   Web of Science ®Google Scholar
• Khavinson D. A note on entire solutions of the eiconal equation. Am Math Mon. 1995;102:159–161.
   Web of Science ®Google Scholar
• Li BQ. On meromorphic solutions of Cn. Math Z. 2008;258:763–771.
   Web of Science ®Google Scholar
• Li BQ. Entire solutions of certain partial differential equations and factorization of partial derivatives. Trans Am Math Soc. 2005;357:3169–3177.
   Web of Science ®Google Scholar
• Saleeby E. Entire and meromorphic solutions of Fermat type partial differential equations. Anal (Munich). 1999;19:369–376.
   Google Scholar
• Saleeby E. Meromorphic solutions of generalized inviscid Burgers’ equations and a family of quadratic PDEs. J Math Anal Appl. 2015;425:508–519.
   Web of Science ®Google Scholar
• Ehrenpreis L. Fourier analysis in several comples variables. New York (NY): Wiley-Interscience; 1970.
   Google Scholar
• Tsuno Y. On a Class of differential operators with polynomial coefficients. Hiroshima Math J. 1973;3:69–79.
   Google Scholar
• Chiang YM, Ismail MEH. On value distribution theory of second order periodic ODEs, special functions and orthogonal polynomials. Can J Math. 2006;58(4):726–767.
   Web of Science ®Google Scholar
• Hu PC, Li BQ. On meromorphic solutions of linear partial differential equations of second order. J Math Anal Appl. 2012;393:200–211.
   Web of Science ®Google Scholar
• Hu PC, Yang CC. A linear homogeneous partial differential equation with entire solutions represented by Bessel polynomials. J Math Anal Appl. 2010;368:263–280.
   Web of Science ®Google Scholar
• Bernstein SN. Sur la nature analytique des solutions de certaines équations aux dériveés parttielles du second order. C R Acad Sci Paris. 1903;137:778–781.
   Google Scholar
• Lewy H. Neuer Beweis des analytischen Charakters der L\"{o}sungen elliptischer Differentialgleichungen. Math Ann. 1929;101:609–619.
   Google Scholar
• Bertran D. Multiplicity and vanishing lemmas for differential and q-difference equations in the Siegel--Shidlovskii theory. J Math Sci. 2012;180:542–549.
   Google Scholar
• Beukers F. A refined version of the Siegel--Shidlovskii theorem. Ann Math. 2006;163:369–379.
   Web of Science ®Google Scholar
• Dwork BM, Gerotto G, and Sullivan FJ . An introduction to G-functions. Princeton (NJ): Princeton University Press; 1994.
   Google Scholar
• Ehrenpreis L. Transcendental numbers and partial differential equations. Lecture notes in mathematics. Berlin: Springer; 1985. (Lecture notes in mathematics, Vol. 1135, p. 112–115).
   Google Scholar
• Szegö G. Orthogonal polynomials. 4th ed. Vol. XXIII. Providence (RI): American Mathematical Society, Colloquium Publications; 1975.
   Google Scholar
• Askey R. Orthogonal polynomials and special functions. Philadelphia (PA): Society for Industrial and Applied Mathematics; 1975.
   Google Scholar
• Askey R. Summability of Jacobi series. Trans Am Math Soc. 1973;179:71–84.
   Web of Science ®Google Scholar
• Bavinck H. Jacobi series and approximation. Mathematical centre tracts, no. 39. Amsterdam: Mathematisch Centrum; 1972.
   Google Scholar
• Hu PC, Li P, and Yang CC. Unicity of meromorphic mappings. Beijing: Kluwer Academic Publishers; 2003.
   Google Scholar
• Laine I. Nevalinna theory and complex differential equations. New York (NY): Walter de Gruyter; 1993.
   Google Scholar
• Stoll W. Holomorphic functions of finite orders in several complex variables, Conference board of the mathematical science. AMS; 1974. (Regional conference series in mathematics, Vol. 21).
   Google Scholar
• Hu PC, Li BQ. Linear partial differential equations of second order associated with orthogonal polynomials. preprint.
   Google Scholar
• Titchmarsh EC. The theory of functions. 2nd ed. Oxford: Oxford University Press; 1968.
   Google Scholar
• Carlson BC. Expansion of analytic functions in Jacobi series. SIAM J Math Anal. 1974;5:797–808.
   Web of Science ®Google Scholar
• Rusev P. Expansion of analytic functions in series of Jacobi polynomials. Bull Inst Math Acad Bulg Sci. 1963;7:61–73.
   Google Scholar
• Rusev P. Expansion of analytic functions in series of classical orthogonal polynomials. Banach Center Publications. 1983;11:287–298.
   Google Scholar
• Whittaker RE, Watson GN. A course of modern analysis. Cambridge: Cambridge University Press; 1963.
   Google Scholar
• Brosch G. Eindeutigkeitssätze für meromorphe Funktionen [dissertation]. Technical University of Aachen; 1989.
   Google Scholar
• Bank SB, Kaufman R. On the growth of meromorphic solutions of the differential equation Cn. Acta Math. 1980;144:233–248.
   Web of Science ®Google Scholar
• Berenstein CA, Chang DC, Li BQ. The uniqueness problem and meromorphic solutions of partial differential equations. J D’Analyse Math. 1999;77:51–68.
   Web of Science ®Google Scholar
• Hu PC, Li BQ. On meromorphic solutions of nonlinear partial differential equations of first order. J Math Anal Appl. 2011;377:881–888.
   Web of Science ®Google Scholar
• Hu PC, Li BQ. A note on meromorphic solutions of linear partial differential equations of second order. Complex Anal Oper Theory. 2014;8:1173–1182.
   Web of Science ®Google Scholar
• Krall HL, Sheffer IM. Orthogonal polynomials in two variables. Ann Mat Pura Appl. 1967;76(4):325–376.
   Google Scholar
• Li BQ. On Fermat-type functional and partial differential equations. The mathematical legacy of Leon Ehrenpreis. Springer; 2012. (Springer proceedings in mathematics, Vol. 16, p. 209–222).
   Google Scholar
• Li BQ. A logarithmic derivative lemma in several complex variables and its applications. Trans Am Math Soc. 2011;363:6257–6267.
   Web of Science ®Google Scholar
• Hayman WK. Meromorphic functions. Oxford: Clarendon Press; 1975.
   Google Scholar
• Li BQ. On reduction of functional-differential equations. Comlex Variables Theory Appl. 1996;31:311–324.
   Google Scholar
• Mokhon’ko AZ, Mokhon’ko VD. Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their applications to differential equations. Sibirsk Mat Zh. 1974;15:1305–1322.
   Google Scholar
• Li BQ. Uniqueness of entie functions sharing four small functins. Am J Math. 1997;119:841–858.
   Web of Science ®Google Scholar
• Courant R, Hilbert D. Methods of mathematical physics. Vol. II, Partial differential equations. New York (NY): Interscience; 1962.
   Google Scholar