Stern polynomials and algebraic independence

Abstract: Let $t\geq2$ and $k\geq1$ be integers. Let $H_{k}(z)$ with $\left\vert z\right\vert <1$ be the limit of a certain subsequence of the Stern polynomials introduced by Dilcher and Eriksen. We use Mahler's method to prove the algebraic independence of the values at nonzero algebraic points of the functions $H_{k}(z)$ and $H_{k}(z^{t^{k}})$.

• The paper shows that for any algebraic α with 0 < |α| < 1, Hₖ(α) and Hₖ(α^(tᵏ)) are algebraically independent.
• The paper employs Mahler's transcendence method, utilizing functional equations and matrix recurrences to rigorously prove the independence.
• The paper extends the analysis to related continued fraction expansions, demonstrating their transcendence and potential applications in Diophantine approximation.

Stern Polynomials and Algebraic Independence

Introduction and Problem Setting

This paper investigates the algebraic independence of certain analytic functions derived from the class of Stern polynomials, extending concepts connected with Stern's diatomic sequence through the mechanism of functional equations and continued fractions. Focusing on the generalization introduced by Dilcher and Eriksen, the authors analyze the power series $H_k(z)$ obtained as the limit of a specific subsequence of these polynomials, defined for integer parameters $t\geq2$, $k\geq1$, and $|z|<1$, and examine the algebraic independence of their values at algebraic points.

Let $a_t(n;z)$ denote the "Type 1" Stern polynomials given recursively by: $a_t(0;z) = 0, \quad a_t(1;z) = 1,$

$$a_t(2n;z) = z\, a_t(n;z^t), \quad a_t(2n+1;z) = a_t(n+1;z^t) + a_t(n;z^t),$$

with coefficients only in $\{0,1\}$. The analytic function of interest $H_k(z)$ arises as a limit of a sequence $\left(a_t(\alpha_n;z)\right)_{n\geq0}$ along a structured subsequence of indices and satisfies a key nonlinear functional equation.

Main Contributions

The core result establishes that for any nonzero algebraic number $t\geq2$0 with $t\geq2$1, the two numbers $t\geq2$2 and $t\geq2$3 are algebraically independent.

This is formally captured as follows:

Theorem: For any algebraic $t\geq2$4 with $t\geq2$5, $t\geq2$6 and $t\geq2$7 are algebraically independent.

The proof is an application of Mahler's method for transcendence and algebraic independence, leveraging the specific structure of the functional equations satisfied by $t\geq2$8 and its iterate.

As a corollary, the values constructed by infinite continued fractions built from the coefficients of these Stern polynomials are shown to be transcendental for the same class of $t\geq2$9.

Methods and Technical Approach

The analysis is predicated on several technical foundations:

• Functional Equation Structure: The function $k\geq1$0 satisfies a Mahler-type functional equation of the form

$k\geq1$1

where $k\geq1$2 and $k\geq1$3 can be explicitly described.

• Matrix Recurrence: The two-dimensional vector of functions $k\geq1$4 admits a linear recursion with matrix coefficients, so the machinery of Mahler's transcendence method applies.
• Continued Fraction Expansion: The analytic expansion of $k\geq1$5 as an infinite continued fraction, where every numerator and denominator is a simple monomial or sum thereof, facilitates identification of irrationality and transcendence via classical and modern results.
• Algebraic Independence Mechanism: By establishing functional independence in the ring of power series with coefficients in $k\geq1$6 (invoking Fatou’s theorem and the explicit characterization of these recurrences), the algebraic independence in values at algebraic arguments follows from Mahler's method results (as formulated by Ku. Nishioka).

Key Results: Numerical and Structural Highlights

• The paper explicitly identifies cases (for various $k\geq1$7) of transcendental continued fractions, e.g., for $k\geq1$8,

$k\geq1$9

and proves that these values are transcendental for algebraic $|z|<1$0 in the open unit disk.

• Strong claims rely on the fact that the coefficients of $|z|<1$1 are Boolean, so analytic rationality would imply algebraicity except for power series proven to be transcendental (via Fatou’s theorem).
• The algebraic independence of $|z|<1$2 and $|z|<1$3 over $|z|<1$4 is demonstrated for all choices of $|z|<1$5, $|z|<1$6 under the specified conditions, without additional side constraints.

Implications and Future Research Directions

From a theoretical perspective, these results contribute to the growing body of work connecting Mahler's classification of transcendental functions with combinatorial and automata-theoretic power series. The extension to algebraic independence (not just transcendence) is nontrivial and adds new cases to the catalog of functions whose special values are "maximally non-algebraic," in contrast to classical hypergeometric and other special functions.

On the practical side, these findings reinforce the power of Mahler’s method for both qualitative (transcendence) and quantitative (algebraic independence) study of sequences and their analytic continuations. Results concerning explicit continued fractions of this diagonal type may have indirect applications in Diophantine approximation and in the analysis of automatic sequences over finite fields.

Further developments may include:

• Extension of such independence results for more general families of recurrences and higher-dimensional systems,
• Investigation of the arithmetic properties when the coefficients are taken from larger finite sets or non-binary alphabets,
• Potential links to automata theory and model-theoretic algebraic independence in the context of formal languages.

Conclusion

This work rigorously establishes the algebraic independence of two analytic functions tied to the limiting behavior of Stern polynomials, for all algebraic values in the open unit disk. The results presented rely on and extend Mahler's framework for transcendence and independence, illuminating new classes of power series whose values escape all algebraic relations at broad swathes of algebraic points. The constructive nature of the argument and the explicit nature of the functions under consideration suggest further generalizations are plausible, and the interplay between continued fraction representations and transcendence methods remains a prolific arena for future mathematical inquiry.