Let Δ d ( a , b ) ( n ) = q d ( a ) ( n ) - Q d ( b ) ( n ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$arDelta _d^{(a,b)}(n) = q_d^{(a)}(n) - Q_d^{(b)}(n)$$end{document} where q d ( a ) ( n ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$q_d^{(a)}(n)$$end{document} counts the number of partitions of n into parts with difference at least d and size at least a , and Q d ( b ) ( n ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Q_d^{(b)}(n)$$end{document} counts the number of partitions into parts ≡ ± b mod d + 3 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$equiv pm b left( mathrm {mod} d + 3ight) $$end{document} . In 1956, Alder conjectured that Δ d ( 1 , 1 ) ( n ) ≥ 0 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$arDelta _d^{(1,1)}(n) ge 0$$end{document} for all positive n and d . This conjecture was proved partially by Andrews in 1971, by Yee in 2008, and was fully resolved by Alfes, Jameson and Lemke Oliver in 2011. Alder’s conjecture generalizes several well-known partition identities, including Euler’s theorem that the number of partitions of n into odd parts equals the number of those into distinct parts, as well as the first of the famous Rogers–Ramanujan identities. In 2020, Kang and Park constructed an extension of Alder’s conjecture which relates to the second Rogers–Ramanujan identity by considering the difference Δ d ( a , b , - ) ( n ) = q d ( a ) ( n ) - Q d ( b , - ) ( n ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$arDelta _d^{(a,b,-)}(n) = q_d^{(a)}(n) - Q_d^{(b,-)}(n)$$end{document} , where Q d ( b , - ) ( n ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Q_d^{(b,-)}(n)$$end{document} counts the number of partitions into parts ≡ ± b mod d + 3 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$equiv pm b left( mathrm {mod} d + 3ight) $$end{document} excluding the part d + 3 - b documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$d+3-b$$end{document} . Kang and Park conjectured that Δ d ( 2 , 2 , - ) ( n ) ≥ 0 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$arDelta _d^{(2,2,-)}(n)ge 0$$end{document} for all d ≥ 1 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$dge 1$$end{document} and n ≥ 0 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$nge 0$$end{document} , and proved this when d = 2 r - 2 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$d = 2^r - 2$$end{document} and n is even. Here, we prove Kang and Park’s conjecture for all d ≥ 62 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setle
展开▼
