کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
5775362 | 1631604 | 2018 | 22 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
Finite difference of the overpartition function
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
ریاضیات کاربردی
پیش نمایش صفحه اول مقاله

چکیده انگلیسی
Let p(n) denote the integer partition function. Good conjectured that Îrp(n) alternates in sign up to a certain value n=n(r), and then it stays positive. Gupta showed that for any given r and sufficiently large n, Îrp(n)>0. Odlyzko proved this conjecture and gave an asymptotic formula for n(r). Then, Almkvist, Knessel and Keller gave many contributions for the exact value of n(r). For the finite difference of logâ¡p(n), DeSalvo and Pak proved that 0â¤ââ³2logâ¡p(nâ1)â¤logâ¡(1+1n) and conjectured a sharper upper bound for ââ³2logâ¡p(n). Chen, Wang and Xie proved this conjecture and showed the positivity of (â1)râ1â³rlogâ¡p(n), and further gave an upper bound for (â1)râ1â³rlogâ¡p(n). As for the overpartition function pâ¾(n), Engel recently proved that pâ¾(n) is log-concave for nâ¥2, that is, ââ³2logâ¡pâ¾(n)â¥0 for nâ¥2. Motivated by these results, in this paper we will prove the positivity of finite differences of the overpartition function and give an upper bound for â³rpâ¾(n). Then we show that for any given râ¥1, there exists a positive number n(r) such that (â1)râ1â³rlogâ¡pâ¾(n)>0 for n>n(r), where â³ is the difference operator with respect to n. Moreover, we give an upper bound for (â1)râ1â³rlogâ¡pâ¾(n).
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Advances in Applied Mathematics - Volume 92, January 2018, Pages 51-72
Journal: Advances in Applied Mathematics - Volume 92, January 2018, Pages 51-72
نویسندگان
Larry X.W. Wang, Gary Y.B. Xie, Andy Q. Zhang,