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Certain quantities transcendental over \(\text{GF}(p^n,x)\). II. (English) Zbl 0063.08102

Author’s introduction: Let \(\text{GF}(q)\) denote a fixed finite field of order \(q=p^n\). Let \(x\) be an indeterminate over \(\text{GF}(q)\) and denote by \(\text{GF}[q, x]\) the ring of polynomials in \(x\) with coefficients from the finite field. \(\text{GF}(q, x)\) will be the quotient field of \(\text{GF}[q, x]\). We are concerned here with the transcendence of certain quantities over \(\text{GF}(q, x)\).
Place \[ [k]=x^{q^k}-x,\quad F_k=[k][k-1]^q\cdots [1]^{q^{k-1}}, \]
\[ F_0 = 1,\quad L_k=[k]\cdots [1],\quad L_0=1. \]
L. Carlitz [Duke Math. J. 1, 137–168 (1935; Zbl 0012.04904 and JFM 61.0127.01)] has studied the function
\[ \psi(t)=\sum_{i=0}^\infty (-1)^i\frac{t^{q^i}}{F_i} \]
and its inverse \[ \lambda=\sum_{i=0}^\infty\frac{t^{q^i}}{L_i}. \]
In particular, there is a quantity \(\xi\neq 0\) (in a suitable field containing \(\text{GF}(q, x)\)) such that \(\psi(E\xi)=0\) for all polynomials \(E\), i.e., all elements of \(\text{GF}[q, x]\). It was proved in a previous paper [Duke Math. J. 8, 701–720 (1941; Zbl 0063.08101)] that if \(\alpha\neq 0\) is algebraic over \(\text{GF}(q, x)\), then \(\psi(\alpha)\) is transcendental. In particular, \(\xi\) is transcendental.
Here we shall prove the transcendence of \[ \sum_{i=0}^\infty\frac 1{L_i^{\gamma}} \] when \(\gamma\) is a positive rational integer. This will enable us to give a new proof of the transcendence of \(\xi\). The theorem could be generalized slightly and similar theorems proved by the same method.
(revised)

MSC:

11T06 Polynomials over finite fields
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