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A q-ANALOG OF RESTRICTED GROWTH FUNCTIONS, DOBINSKI'S EQUALITY, AND CHARLIER POLYNOMIALS

Analysis · math.CO · math.CV · posed by Stephen C. Milne · open

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Statement

Moreover, we conjecture that the only finite singularities of Φq(t)\Phi_{q}(t) are of the form qm/(q−1),m≥1.q^{m}/(q-1),\quad m \geq 1.

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  • A q-ANALOG OF RESTRICTED GROWTH FUNCTIONS, DOBINSKI'S EQUALITY, AND CHARLIER POLYNOMIALS
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
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    — the result was found by a model.

    NEW

    Problem: For fixed real q>1q>1, put

    [n]q=1+q+⋯+qn−1,[n]q!=∏r=1n[r]q,[n]_q=1+q+\cdots+q^{n-1},\qquad [n]_q!=\prod_{r=1}^n [r]_q,

    and let Bn(q)B_n(q) be Milne’s qq-Bell numbers, equivalently given by the qq-Dobinski formula

    Bn(q)=1eq(1)∑j≥0[j]qn[j]q!,eq(z)=∑n≥0zn[n]q!.B_n(q)=\frac1{e_q(1)}\sum_{j\ge0}\frac{[j]_q^n}{[j]_q!},\qquad e_q(z)=\sum_{n\ge0}\frac{z^n}{[n]_q!}.

    Milne’s generating function is

    Φq(t)=∑n≥0Bn(q)tn[n]q!.\Phi_q(t)=\sum_{n\ge0}B_n(q)\frac{t^n}{[n]_q!}.

    The conjecture is that the only finite singularities of Φq(t)\Phi_q(t) are

    qmq−1,m≥1.\frac{q^m}{q-1},\qquad m\ge1.

    The hypothesis q>1q>1 is the natural analytic one: for q=1q=1 the displayed points are undefined, and for 0<q<10<q<1 they accumulate at 00.

    Result: The conjecture is true; in fact the singularities are exactly those points, and the point qm/(q−1)q^m/(q-1) is a pole of order mm.

    Proof. Set Q=q−1∈(0,1)Q=q^{-1}\in(0,1), x=(1−Q)tx=(1-Q)t, and use (a;Q)n=∏r=0n−1(1−aQr)(a;Q)_n=\prod_{r=0}^{n-1}(1-aQ^r). Euler’s product gives

    eq(z)=(−(1−Q)z;Q)∞.e_q(z)=(-(1-Q)z;Q)_\infty .

    By qq-Dobinski,

    Φq(t)=1eq(1)∑j≥0eq([j]qt)[j]q!.\Phi_q(t)=\frac1{e_q(1)}\sum_{j\ge0}\frac{e_q([j]_q t)}{[j]_q!}.

    The jj-th summand is

    Tj(t)=eq([j]qt)[j]q!.T_j(t)=\frac{e_q([j]_q t)}{[j]_q!}.

    Since

    [j]q!=Q−j(j−1)/2(Q;Q)j(1−Q)j[j]_q!=\frac{Q^{-j(j-1)/2}(Q;Q)_j}{(1-Q)^j}

    and (1−Q)[j]q=Q1−j(1−Qj)(1-Q)[j]_q=Q^{1-j}(1-Q^j), one obtains, for large jj,

    Tj(t)=xj(1−Qj)jH(t,Qj),T_j(t)=x^j(1-Q^j)^j H(t,Q^j),

    where

    H(t,u)=(uQ;Q)∞(Q;Q)∞(−1/(t(1−u));Q)∞(−u/(t(1−u));Q)∞(−tQ(1−u);Q)∞.H(t,u)= \frac{(uQ;Q)_\infty}{(Q;Q)_\infty} \frac{(-1/(t(1-u));Q)_\infty}{(-u/(t(1-u));Q)_\infty} (-tQ(1-u);Q)_\infty .

    On every compact subset of C×\mathbb C^\times, H(t,u)H(t,u) is holomorphic for ∣u∣|u| sufficiently small and has a power expansion

    H(t,u)=∑r=0Nhr(t)ur+O(uN+1).H(t,u)=\sum_{r=0}^N h_r(t)u^r+O(u^{N+1}).

    Also

    (1−u)j=∑ℓ=0N(−1)ℓ(jℓ)uℓ+O(jN+1∣u∣N+1).(1-u)^j=\sum_{\ell=0}^N(-1)^\ell\binom{j}{\ell}u^\ell+O(j^{N+1}|u|^{N+1}).

    With u=Qju=Q^j, this gives

    Tj(t)=∑n=0NPn(j,t)(xQn)j+O ⁣(jN+1∣x∣jQ(N+1)j),T_j(t)=\sum_{n=0}^N P_n(j,t)(xQ^n)^j +O\!\left(j^{N+1}|x|^j Q^{(N+1)j}\right),

    where each Pn(j,t)P_n(j,t) is a polynomial in jj with holomorphic coefficients in tt.

    Choose NN so large that ∣x∣QN+1<1|x|Q^{N+1}<1 on a given compact set. The error then sums normally, while

    ∑j≥Jjr(xQn)j\sum_{j\ge J} j^r(xQ^n)^j

    is a rational function with possible pole only at xQn=1xQ^n=1. Hence Φq\Phi_q extends meromorphically to C\mathbb C, with possible finite poles only when

    xQn=1⟺t=Q−n1−Q=qn+1q−1.xQ^n=1 \quad\Longleftrightarrow\quad t=\frac{Q^{-n}}{1-Q}=\frac{q^{n+1}}{q-1}.

    Thus there are no other finite singularities.

    Finally, at tm=qm/(q−1)t_m=q^m/(q-1), with n=m−1n=m-1, the highest-degree part of Pn(j,t)P_n(j,t) is

    (−1)nh0(t)n!jn,(-1)^n\frac{h_0(t)}{n!}j^n,

    and

    h0(t)=H(t,0)=(−1/t;Q)∞(−tQ;Q)∞(Q;Q)∞.h_0(t)=H(t,0)=\frac{(-1/t;Q)_\infty(-tQ;Q)_\infty}{(Q;Q)_\infty}.

    For t=tm>0t=t_m>0, this is nonzero. Therefore the contribution

    ∑jPn(j,t)(xQn)j\sum_j P_n(j,t)(xQ^n)^j

    has a genuine pole of order n+1=mn+1=m. Hence the finite singularities are exactly

    qmq−1,m≥1.\frac{q^m}{q-1},\qquad m\ge1.

    Citation: Milne’s paper is the source of the definitions, qq-Dobinski formula, and conjecture: Stephen C. Milne, “A qq-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.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE2

    PASS

    The proof attacks the correct conjecture (for the intended analytic regime q>1q>1) and is mathematically sound. It rewrites Milne’s qq-Dobinski series, obtains a uniform asymptotic expansion of the summands in powers of QjQ^j, and sums the resulting polynomial-geometric tails to get a meromorphic continuation with possible poles only at t=qm/(q−1)t=q^m/(q-1). 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 mm. 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 qq-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”, “qm/(q−1)q^m/(q-1)”, and the exact Milne title led back to Milne’s paper, general qq-Bell/qq-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 qq-Poisson/qq-Dobinski and Johnson on alternative qq-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 qq-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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