A Difference Picard Theorem for Meromorphic Functions of Several Variables

作者：Korhonen Risto^*

来源：Computational Methods and Function Theory, 2012, 12(1): 343-361.

DOI：10.1007/BF03321831

• Picard%26apos
• s Theorem
• Second Main Theorem
• partial difference equation
• several variables
• difference analogue

It is shown that if n is an element of N, c is an element of C-n, and three distinct values of a meromorphic function f : C-n -%26gt; P-1 of hyper-order c(f) strictly less than 2/3 have forward invariant pre-images with respect to a translation tau: C-n -%26gt; C-n, tau(z) = z + c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green%26apos;s Picard-Type Theorem in the special case where c(f) %26lt; 2/3, since the empty pre-images of the usual Picard exceptional values are by de finition always forward invariant. In addition, difference analogues of the Lemma on the Logarithmic Derivative and of the Second Main Theorem of Nevanlinna theory for meromorphic functions C-n -%26gt; P-1 are given, and their applications to partial difference equations are discussed.

• 出版日期2012