Lattice paths and positive trigonometric sums
The authors prove that if \(| M-N| \leq k\), then the expansion of \(\sum_l \left( M+N \atop M-kl \right) \cos(lx)\), expanded as a power series in \(1 + \cos(x)\), has non-negative integer coefficients which can be interpreted as counting certain lattice paths. This problem arose in connection with a conjecture posed by Peter Borwein which is equivalent to the statement that the polynomial in \(q\): \[ \sum_l \left[ M+N \atop M-lk \right] (-1)^l q^{(l^2k(a+b)+lk(b-a))/2} \] has non-negative coefficients when \(k=3\) and \((a,b) = (2/3,7/3)\), \((5/3,4/3)\), or \((8/3,1/3)\) [\textit{G. E. Andrews}, J. Symb. Comput. 20, No. 5-6, 487-501 (1995; Zbl 0849.68062)]. When \(k\), \(a\), and \(b\) are integers, these coefficients count partitions and so are non-negative. The authors prove that the coefficients are non-negative also when \(k\) even, \(M=N\), and \(a = (k+1)/2 = b+1\) or \(a = b = k/2\) and when \(M+N\) is even, \(a+b=k\), and \(k-2a = N-M\pm 1\).
scientific article; zbMATH DE number 1271883