Let F=X^2-PX+Q be the characteristic polynomial of a linear sequence (T_n). For any r≥ 0, define polynomials f_r,j(P,Q), for j=0,…,2r+2, by F^r+1=∑_j=0^2r+2f_r,j(P,Q)X^j. Recurrence conjecture. The sequence (tau_n,r) satisfies the linear recurrence of order 2r+2 ∑_j=0^2r+2f_r,j(P,Q)tau_n+j,r=0. The conjecture is motivated by the explicitly verified recurrences for (tau_n,1) and (tau_n,2). It proposes a uniform recurrence for every r≥ 0, but the supplied text does not indicate whether the claim has been proved or remains open.

Recurrence conjecture for the Maclaurin-series coefficients of Somos-related sequences

About 16 years old · traced to 2010

Let $F=X^2-PX+Q$ be the characteristic polynomial of a linear sequence $(T_n)$ . For any $r\geq 0$ , define polynomials $f_{r,j}(P,Q)$ , for $j=0,\ldots,2r+2$ , by

F^{r+1}=\sum_{j=0}^{2r+2}f_{r,j}(P,Q)X^j.

Recurrence conjecture. The sequence $(\tau_{n,r})$ satisfies the linear recurrence of order $2r+2$

\sum_{j=0}^{2r+2}f_{r,j}(P,Q)\tau_{n+j,r}=0.

The conjecture is motivated by the explicitly verified recurrences for $(\tau_{n,1})$ and $(\tau_{n,2})$ . It proposes a uniform recurrence for every $r\geq 0$ , but the supplied text does not indicate whether the claim has been proved or remains open.

References

Primary source

Andrei K. Svinin, “Somos-4 equation and related equations”, arXiv:2307.05866 (2023).

Additional references

4 papers in this index state this conjecture (2010–2023). The statement above is taken from the most recent of them; the others are arXiv:2210.10577, arXiv:1910.09576, arXiv:1002.3844.

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Open

No public discussion or published progress on this conjecture was found.

No public discussion or published progress addressing this conjecture was found.

Current status (as of August 2026): The conjecture appears open, with no recorded public activity establishing or refuting it.

ProofThis solution needs a summarySee full solutionHide full solution

Let E denote the forward shift and put F(E)=E²−PE+Q, with Q≠0 as in the source. The already proved Theorem 7.2 gives τ_n(x)=(T_n−T_{n−1}B(x))e^{nx}, where F(E)T=0, B(0)=0, and B is independent of n. Write B(x)=Σ_{j≥1}b_jx^j/j!, τ_n(x)=Σ_{r≥0}τ_{n,r}x^r/r!. Coefficient extraction yields, for every r≥0, τ_{n,r}=n^rT_n−T_{n−1}Σ_{j=1}^r C(r,j)b_j n^{r−j}.

Both U_n=T_n and V_n=T_{n−1} satisfy F(E)U=F(E)V=0. More generally, if F(E)U=0 and p(n) is a polynomial of degree at most d, write (f₀,f₁,f₂)=(Q,−P,1). For every shift h≥0, F(E)(p(n)E^hU) =Σ_{s=0}² f_s[p(n+s)−p(n)]E^{h+s}U. Indeed the omitted term p(n)E^hF(E)U vanishes. Every polynomial difference on the right has degree at most d−1. Therefore F(E) lowers by at least one the polynomial degree in the span of polynomial multiples of shifts of U. Consequently F(E)^{d+1}(p(n)U_n)=0. The same argument applies to V.

Since the displayed formula for τ_{n,r} is a sum of polynomial multiples of U_n and V_n with degrees at most r, it follows that F(E)^{r+1}τ_{n,r}=0 for every r≥0. Writing F(X)^{r+1}=Σ_{j=0}^{2r+2}f_{r,j}(P,Q)X^j gives exactly Σ_{j=0}^{2r+2} f_{r,j}(P,Q)τ_{n+j,r}=0. Thus the conjectured recurrence holds in every order, including repeated characteristic roots. In fact the argument works for any formal series B(x), without requiring the source's additional Riccati equation.