bibkey: parisse2024hypersequences authors: Daniele Parisse year: 2024 title: “On Hypersequences of an Arbitrary Sequence and Their Weighted Sums” doi: 10.5281/zenodo.13331499 url: https://math.colgate.edu/~integers/y70/y70.pdf claim: “Section 4 states two conjectures: the alternating factorial-Stirling weighted sum of the even-index Fibonacci numbers equals the signed plain weighted sum of the preceding Stirling row, and the same identity with the Lucas numbers in place of the Fibonacci numbers.” strata_touched:
- D5/S3/Combinatorics/ParisseStirlingFibonacciAlternatingSum license: citation-only triage: anchor
Parisse, hypersequences of an arbitrary sequence and their weighted sums
Verified locator
DOI: 10.5281/zenodo.13331499
URL: https://math.colgate.edu/~integers/y70/y70.pdf
Integers 24 (2024), Article A70, received 8/28/23, revised 9/28/23, accepted 7/22/24, published 8/16/24. The author is Daniele Parisse, Airbus Defence and Space GmbH, Manching, Germany. The volume index at https://math.colgate.edu/~integers/vol24.html carries the same author, title and article number, and the DOI printed on page 2 of the article resolves to the same record.
Definitions
For a sequence (a_n) the article writes a_n^{(r)} for the hypersequence of the
r-th generation, defined by a_n^{(r)} = ∑_{k=0}^{n} a_k^{(r-1)} with
a_n^{(0)} = a_n, and studies the weighted sums
t_ℓ^{(r)}(n) = ∑_{k=0}^{n} k^ℓ a_k^{(r)}. Theorem 9 expresses these through the
coefficients
c_{ℓ,m}(n) = ∑_{k=0}^{m} (-1)^k C(m, k) (k + n + 1)^ℓ ,
which do not depend on r. Section 3 writes S(j, m) for the Stirling numbers of
the second kind, defined there as the number of partitions of a set of j
elements into exactly m nonempty blocks, and Equation (3.14) evaluates the
constant term
c_{ℓ,m}(0) = (-1)^m m! S(ℓ + 1, m + 1) .
F denotes the Fibonacci numbers with F_1 = F_2 = 1 and L the Lucas numbers
with L_0 = 2, L_1 = 1.
The statements in question
Section 4 applies Theorem 9 to the Fibonacci sequence, obtaining Equation (4.7), and then states:
Conjecture 1. For all ℓ ∈ N_0, we have
∑_{m=0}^{ℓ} (-1)^m m! S(ℓ+1, m+1) F_{2(m+1)} = (-1)^ℓ ∑_{m=0}^{ℓ} m! S(ℓ, m) F_{m+2} .
The remark immediately after reads: “Note that the unsigned sum on the
right-hand side (the sequence A000557 in [21]) is exactly the sequence M_{2,j}.”
That remark fixes the reading of the stacked bracket symbols as Stirling numbers
of the second kind rather than binomial coefficients, because A000557 is recorded
as a(n) = ∑_{k=0..n} k! Stirling2(n, k) Fibonacci(k+2).
The Lucas analogue follows the same pattern. After Equation (4.9) the article states:
Conjecture 2. For all ℓ ∈ N_0, we have
∑_{m=0}^{ℓ} (-1)^m m! S(ℓ+1, m+1) L_{2(m+1)} = (-1)^ℓ ∑_{m=0}^{ℓ} m! S(ℓ, m) L_{m+2} .
with the remark that multiplying the right-hand side by -1 gives A263968, whose
recorded formula is a(n) = (-1)^{n+1} ∑_{k=0..n} k! Lucas(k+2) Stirling2(n, k).
Scope of the recorded answer
Both identities hold, and they hold for the same reason: neither depends on the
initial values of the sequence, only on the recurrence u_{n+2} = u_n + u_{n+1}.
The recorded result establishes the common generalisation for an arbitrary
integer sequence satisfying that recurrence and reads off the two conjectures as
the Fibonacci and Lucas instances.
Bounded prior-resolution evidence
The author’s own sequel, On Hypersequences of an Arbitrary Sequence and Their Weighted Sums II, Integers 26 (2026), Article A36, published 2/20/26, was opened in full: it contains no occurrence of the word conjecture and does not return to either statement. The OEIS entries A000557 and A263968 were opened; both link this article and neither records the alternating identity among its formulas or any proof of it. A web pass over the two sequence numbers and the article title returned no proof; the nearest recent item, arXiv:2511.10797 on weighted sums of Lucas sequences, mentions neither Parisse nor Stirling numbers. Citation-index result pages were not exhaustively reachable, so this is a bounded negative finding and no worldwide priority claim is made.