The Linear Difference Equation of the First Order

Abstract

Resume.-2.Reduction of the General Difference Equation tp{z)f(z + u)-x(?)f(z) = ik) to Two Standard Equations.-3.Definitions.-4-7.Solution of /(z + eu) = </>(z)/(z), where <f>(z) is holomorphic, of finite order, without extraneous exponential factor, with zeros all negative with respect to a>.-8.Solution of /(z + a) = <f> (z)f(z), where <f> («) is holomorphic, of finite order, without extraneous exponential factor, the zeros being positive with respect to a.-9, 10.Solution of f(z + w) = fi(z)f{z), where /M(Z) is meromorphic and possibly of infinite order.-11,12.The Riemann ( Function and its Modification.Solution of f(z + w)-f(z) = G(z), where G{:) is an integral function.-13.Completion of $10.-14, 15.Other Forms for the Solution of / ( s + &>) = n(z)f(z).-16.Extension to Cases where M(Z) haa Repeated Zeros or Poles or is of Multiple Sequence.-17-22.Solution of / ( s + «) -/(«) = ju( z ) "* Three Different Forms.-23.The Nature of the Solution of <p(z)f(z + to)x {z)/(z) = t|/(s).-24.Examples.Conclusion.

Locations

• Proceedings of the London Mathematical Society - View
• Zenodo (CERN European Organization for Nuclear Research) - View - PDF