[Paper Review] Higher Order Tur\'an Inequalities for the Partition Function

William Y. C. Chen, Dennis X. Q. Jia+1 more|arXiv (Cornell University)|Jun 30, 2017

Advanced Mathematical Identities14 references3 citations

TL;DR

This paper proves that the partition function $ p(n) $ satisfies higher order Turán inequalities for $ n \geq 95 $, using the Hardy-Ramanujan-Rademacher formula to derive tight bounds on the ratio $ p(n+1)p(n-1)/p(n)^2 $. The key result confirms that the associated Jensen polynomials $ g_{3,n-1}(x) $ have only real roots for $ n \geq 95 $, supporting a broader conjecture on real-rootedness of higher-degree Jensen polynomials for $ p(n) $.

ABSTRACT

The Tur\\'{a}n inequalities and the higher order Tur\\'{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P\\'{o}lya class. A real sequence $\\{a_{n}\\}$ is said to satisfy the Tur\\'{a}n inequalities if for $n\\geq 1$, $a_n^2-a_{n-1}a_{n+1}\\geq 0$. It is said to satisfy the higher order Tur\\'{a}n inequalities if for $n\\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\\geq 0$. A sequence satisfying the Tur\\'an inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence $\\{ p(n)\\}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by Chen that $p(n)$ satisfies the higher order Tur\\'{a}n inequalities for $n\\geq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $n\\geq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $m\\geq 4$ there exists an integer $N(m)$ such that for $n\\geq N(m) $, the polynomials $\\sum_{k=0}^m {m\\choose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono.

Motivation & Objective

• To prove Chen's conjecture that the partition function $ p(n) $ satisfies higher order Turán inequalities for $ n \geq 95 $.
• To establish that the Jensen polynomials $ g_{3,n-1}(x) = p(n-1) + 3p(n)x + 3p(n+1)x^2 + p(n+2)x^3 $ have only real roots for $ n \geq 95 $.
• To provide a rigorous analysis of the ratio $ u_n = p(n+1)p(n-1)/p(n)^2 $ using asymptotic expansions from the Hardy-Ramanujan-Rademacher formula.
• To support a broader conjecture that for any $ m \geq 4 $, the polynomials $ \sum_{k=0}^m \binom{m}{k} p(n+k) x^k $ have only real roots for sufficiently large $ n $.
• To confirm the validity of intermediate inequalities involving $ H(x) $, $ G(x) $, and $ \psi(t) $ that underpin the main proof.

Proposed method

• Derives upper and lower bounds for $ p(n+1)p(n-1)/p(n)^2 $ using the Hardy-Ramanujan-Rademacher asymptotic formula for the partition function.
• Introduces auxiliary functions $ G(x) $ and $ H(x) $ to model the asymptotic behavior of the partition function ratio, enabling comparison via rational approximations.
• Employs the function $ \psi(t) = \frac{t + \frac{\sqrt{5}+1}{2}}{(\sqrt{1-t}+1)^2(\sqrt{1-t}+t)} $ to analyze the discriminant condition in higher order Turán inequalities.
• Uses polynomial dominance arguments to show that $ 2H(x) - (\sqrt{5}-1)G(x) > 0 $ for $ x \geq 134 $, which implies the required inequality for $ \psi(H(x)/G(x)) $.
• Applies a chain of inequalities involving $ f(n) $, $ g(n) $, and $ \mu(n) $, with $ \mu(n) = n(n-1)(n-2)(n-3)(n-4) $, to bound $ u_n $ and $ Q(u_n) $.
• Verifies the key inequality $ F(u_{n+1}) > 0 $ by showing $ u_n < u_{n+1} < Q(u_n) $ for $ n \geq 1207 $, relying on known monotonicity and asymptotic bounds.

Experimental results

Research questions

• RQ1 Does the partition function $ p(n) $ satisfy the higher order Turán inequalities for all $ n \geq 95 $?
• RQ2 Are the Jensen polynomials $ g_{3,n-1}(x) $ associated with $ p(n) $ real-rooted for $ n \geq 95 $?
• RQ3 Can the ratio $ p(n+1)p(n-1)/p(n)^2 $ be bounded tightly enough to verify the cubic Newton inequality condition?
• RQ4 Is there a uniform bound on $ p(n+1)p(n-1)/p(n)^2 $ that ensures the discriminant of the cubic form remains nonnegative?
• RQ5 Does the broader conjecture hold that for any $ m \geq 4 $, the polynomial $ \sum_{k=0}^m \binom{m}{k} p(n+k) x^k $ has only real roots for sufficiently large $ n $?

Key findings

• The higher order Turán inequality holds for $ p(n) $ when $ n \geq 95 $, as $ 4(1 - u_n)(1 - u_{n+1}) - (1 - u_n u_{n+1})^2 > 0 $, where $ u_n = p(n+1)p(n-1)/p(n)^2 $.
• For $ n \geq 95 $, the Jensen polynomial $ g_{3,n-1}(x) $ has only real roots, confirming a key consequence of the higher order Turán inequality.
• The proof relies on establishing $ u_n < u_{n+1} < Q(u_n) $ for $ n \geq 1207 $, where $ Q(u_n) $ is derived from the cubic discriminant condition.
• The inequality $ \psi(H(x)/G(x)) > (1 - H(x)/G(x))^{3/2} $ holds for $ x \geq 134 $, which is critical for bounding the higher order Turán expression.
• For $ n \geq 35457 $, the bound $ (1 - H(x)/G(x))^{3/2} > 110 / \mu(n-1)^5 $ is verified, where $ \mu(n) = n(n-1)(n-2)(n-3)(n-4) $, supporting the main inequality.
• The final inequality $ x^{10}(G(x) - H(x))^3 - 110^2 G(x)^3 > 0 $ holds for $ x \geq 483 $, confirming the chain of bounds used in the proof.