bibkey: thanatipanonda2014zudilin authors: Thotsaporn Thanatipanonda year: 2014 title: “Beyond Zudilin’s Conjectured q-analog of Schmidt’s problem” doi: 10.1080/10236198.2014.917635 url: https://arxiv.org/abs/1403.4962v1 claim: “Conjecture 4.1: integral linearization of shifted binomial powers with an integer recursion independent of the power index.” strata_touched:

• D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization license: citation-only triage: anchor

Thanatipanonda’s shifted binomial power conjecture

The source is arXiv:1403.4962v1, Section 4, p. 6. The journal citation is Journal of Difference Equations and Applications 20 (2014), 1344–1349.

Conjecture 4.1 reads:

For any integers d, k ≥ 0 and r ≥ 1, there exist integers $a_{d,k,j}^{(r)}$ such that ${\left(\genfrac{}{}{0ex}{}{n+dk}{k}\right)}^r={\sum }_j{a}_{d,k,j}^{(r)}\left(\genfrac{}{}{0ex}{}{n+dj}{j}\right)$ for all n = 0, 1, 2, …. Moreover $a_{d,k,j}^{(r)}$ can be defined as following: $a_{d,k,k}^{(1)}=1$, $a_{d,k,j}^{(1)}=0$ for j ≠ k and $a_{d,k,j}^{(r+1)}={\sum }_i{S}_d(k,j,i)a_{d,k,i}^{(r)}$, where, $S_d(k,j,i)$ are integers, independent of r, for all d, k, j, i.

The natural-number parameters encode all nonnegative integer parameters and all arguments n = 0, 1, 2, …. Binomial values are cast to integers before taking powers. Integer coefficients need not be nonnegative. The finite ranges in the formal statement are supported by the bounds j≤rk for the power coefficients and j≤i+k for the structure constants.

Conjecture 4.2, concerning holonomicity and the absence of a first-order closed form for the structure constants outside d = 0, 1, is separate from Conjecture 4.1. The already-proved Schmidt and q-analog results in Sections 2–3 are also separate.

The arXiv text is the checked source. The journal text has not been read; its retention of the conjecture is ASSUMED-UNVERIFIED.

Verified locator

• arXiv: https://arxiv.org/abs/1403.4962v1, Section 4, Conjecture 4.1, p. 6.
• DOI: https://doi.org/10.1080/10236198.2014.917635. Crossref identifies the journal article and its author; this locator does not attest to the unread journal statement.