Differential Independence of G and ?

• Autores: L. Markus
• Localización: Journal of dynamics and differential equations, ISSN 1040-7294, Vol. 19, Nº. 1, 2007, págs. 133-154
• Idioma: inglés
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• Resumen

  • In a celebrated theorem Hölder proved that the Euler G-function is differential transcendental, i.e. G(z) is not a solution of any (non-trivial) algebraic ordinary differential equation with coefficients that are complex numbers; and we extend his methods to the Riemann ?-function. Moreover, we conjecture that G and ? are differential independent, i.e. G(z) is not a solution of any such algebraic differential equation¿even allowing coefficients that are differential polynomials in ?(z). However, we are able to demonstrate only the partial result that G(z) and ?(sin 2pz) are differential independent.