Sun’s Lowercase Recurrence Sequences

Abstract

The two lowercase polynomial sequences are defined by Sun’s literal initial values and three-term recurrences over the reals.

The natural index starts at zero; x is real. Each recurrence is implemented at successor index n+1 for n at least one. Division by the positive square or cube of n+1 is definitionally equivalent to the multiplied equation displayed below. These are the lowercase g and v, not the older uppercase G and V obtained by a separate substitution.

Definition 1.1 (The lowercase g sequence).

$$\begin{array}{r}g\left(x,0\right)=1\\ g\left(x,1\right)=\frac{x+1}{2}\\ {\left(n+1\right)}^2\cdot g\left(x,n+1\right)=\left(2\cdot n\cdot \left(n+1\right)+\frac{x+1}{2}\right)\cdot g\left(x,n\right)-n^2\cdot g\left(x,n-1\right)\end{array}$$

Formalization. D5/S1/Recurrence/Sun/Sequences.g (✓ std3).

Citation. Zhi-Hong Sun (2026). Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers. DOI: 10.48550/arXiv.2608.13192. URL: https://arxiv.org/html/2608.13192v1.

Commentary.

For every real x, the first values are g(0)=1 and g(1)=(x+1)/2. The displayed equation is the defining recurrence for every n at least one, including its quadratic scale.

Definition 1.2 (The lowercase v sequence).

$$\begin{array}{r}v\left(x,0\right)=1\\ v\left(x,1\right)=x\\ {\left(n+1\right)}^3\cdot v\left(x,n+1\right)=\left(2\cdot n+1\right)\cdot \left(n\cdot \left(n+1\right)+x\right)\cdot v\left(x,n\right)-n^3\cdot v\left(x,n-1\right)\end{array}$$

Formalization. D5/S1/Recurrence/Sun/Sequences.v (✓ std3).

Citation. Zhi-Hong Sun (2026). Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers. DOI: 10.48550/arXiv.2608.13192. URL: https://arxiv.org/html/2608.13192v1.

Commentary.

For every real x, the first values are v(0)=1 and v(1)=x. The displayed equation is the defining recurrence for every n at least one, including its cubic scale.

References

• Truth anchor: D5/S1/Recurrence/Sun/Sequences.g
• Truth anchor: D5/S1/Recurrence/Sun/Sequences.v