2022/07/19 by Bernhard Heim, Markus Neuhäuser, Heim, Bernhard +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2207.09409
openalex publication_date 2022/07/19 · openalex created_date 2022/07/21 · openalex updated_date 2026/07/28
In the 1970s, Nicolas proved that the partition function p(n) is log-concave for n > 25. In \citeHNT21, a precise conjecture on the log-concavity for the plane partition function \funcpp(n) for n >11 was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences \gd(n)\d,n with gd(1)=1 and 0 ≤ gd( n) - nd≤ g1( n) ( n-1) d-1 polynomials \Pngd(x)\d,n given by ∑n=0∞ Pngd(x) qn := \funcexp( x ∑n=1∞ gd(n) (qn)/(n) ) =∏n=1∞ ( 1 - qn )-x fd(n). We recover p(n)= Pnσ1(1) and \funcpp( n) = Pnσ2(1), where σd (n):= ∑ℓ | n ℓd and fd(n)= nd-1. Let n ≥ 6. Then the sequence \Pnσd(1)\d is log-concave for almost all d if and only if n is divisible by 3. Let \funcid(n)=n. Then Pn^\funcid(x) = (x)/(n) Ln-1(1)(-x), where Ln( α) ( x) denotes the α-associated Laguerre polynomial. In this paper, we invest in Turán inequalities Δngd(x) := ( Pngd(x) )2 - Pn-1gd(x) Pn+1gd(x) ≥ 0. Let n ≥ 6 and 0 ≤ x < 2 - (12)/(n+4). Then n is divisible by 3 if and only if Δngd(x) ≥ 0 for almost all d. Let n ≥ 6 and n \not≡ 2 \pmod3. Then the condition on x can be reduced to x ≥ 0. We determine explicit bounds. As an analogue to Nicolas' result, we have for g1= \funcid that Δn^\funcid(x) ≥ 0 for all x ≥ 0 and all n.