A q-ANALOG OF RESTRICTED GROWTH FUNCTIONS, DOBINSKI'S EQUALITY, AND CHARLIER POLYNOMIALS
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Statement
Moreover, we conjecture that the only finite singularities of are of the form
Context
Candidate 3 of the open problems stated in "A q-ANALOG OF RESTRICTED GROWTH FUNCTIONS, DOBINSKI'S EQUALITY, AND CHARLIER POLYNOMIALS", extracted for the Scalable Mathematical Discovery run.
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Problem: For fixed real , put
and let be Milne’s -Bell numbers, equivalently given by the -Dobinski formula
Milne’s generating function is
The conjecture is that the only finite singularities of are
The hypothesis is the natural analytic one: for the displayed points are undefined, and for they accumulate at .
Result: The conjecture is true; in fact the singularities are exactly those points, and the point is a pole of order .
Proof. Set , , and use . Euler’s product gives
By -Dobinski,
The -th summand is
Since
and , one obtains, for large ,
where
On every compact subset of , is holomorphic for sufficiently small and has a power expansion
Also
With , this gives
where each is a polynomial in with holomorphic coefficients in .
Choose so large that on a given compact set. The error then sums normally, while
is a rational function with possible pole only at . Hence extends meromorphically to , with possible finite poles only when
Thus there are no other finite singularities.
Finally, at , with , the highest-degree part of is
and
For , this is nonzero. Therefore the contribution
has a genuine pole of order . Hence the finite singularities are exactly
Citation: Milne’s paper is the source of the definitions, -Dobinski formula, and conjecture: Stephen C. Milne, “A -analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials,” Trans. Amer. Math. Soc. 245 (1978), 89–118, DOI: 10.1090/S0002-9947-1978-0511401-0. The resolution above is the proof given here.
Reviews
0 human reviews · 1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: correctscope Full solution as submitted; SMD novelty classification TYPE2
PASS
The proof attacks the correct conjecture (for the intended analytic regime ) and is mathematically sound. It rewrites Milne’s -Dobinski series, obtains a uniform asymptotic expansion of the summands in powers of , and sums the resulting polynomial-geometric tails to get a meromorphic continuation with possible poles only at . The leading coefficient at each such point is shown nonzero, giving genuine poles of the claimed orders. I found no fatal gap or mismatch.
Novelty assessment
TYPE2
Classification rationale: This appears to be a genuine resolution of an explicit conjecture in Milne’s 1978 TAMS paper, with the added strengthening that the singularities are poles of exact order . The result is specialized and unlikely to interest a top combinatorics journal, but it is more than a routine exercise: it gives a clean meromorphic continuation/pole classification for a named -Bell generating function. It would plausibly support a short standalone note in a standard specialized combinatorics/q-series journal.
Literature check: I found no prior proof of this singularity conjecture. Searches for exact and variant phrases such as “finite singularities” + “q-Bell”, “Milne” + “Phi_q”, “q-Dobinski” + “singularities”, “”, and the exact Milne title led back to Milne’s paper, general -Bell/-Dobinski literature, or unrelated material. arXiv searches for “q-Dobinski”, “q-Bell numbers”, “Phi_q Bell”, and “q-Bell singularities” did not reveal a solution. Later papers/notes found, such as Kwasniewski on -Poisson/-Dobinski and Johnson on alternative -Bell numbers, do not address Milne’s finite-singularity conjecture. OpenAlex identifies the Milne paper and many citations, but I found no citing or related open source containing this theorem.
Citation: Stephen C. Milne, “A -analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials,” Trans. Amer. Math. Soc. 245 (1978), 89–118, DOI: 10.1090/S0002-9947-1978-0511401-8.
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