Positivity of rational functions and their diagonals. (English) Zbl 1374.05023

Straub, Armin; Zudilin, Wadim

J. Approx. Theory 195, 57-69 (2015).

Summary: The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szegő as well as Askey and Gasper, who inspired more recent work. It is well known that the diagonal coefficients of rational functions are \(D\)-finite. This note is motivated by the observation that, for several of the rational functions whose positivity has received special attention, the diagonal terms in fact have arithmetic significance and arise from differential equations that have modular parametrization. In each of these cases, this allows us to conclude that the diagonal is positive.
Further inspired by a result of Gillis, Reznick and Zeilberger, we investigate the relation between positivity of a rational function and the positivity of its diagonal.

MSC:

05A15	Exact enumeration problems, generating functions
33F10	Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)

Keywords:

positivity; rational function; hypergeometric function; modular function; Apéry-like sequence; multivariate asymptotics

Software:

MultiZeilberger

References:

[1]	Almkvist, G.; van Straten, D.; Zudilin, W., Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinb. Math. Soc., 54, 273-295 (2011) · Zbl 1223.33007
[2]	Apagodu, M.; Zeilberger, D., Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory, Adv. Appl. Math., 37, 139-152 (2006) · Zbl 1108.05010
[3]	Askey, R., Certain rational functions whose power series have positive coefficients. II, SIAM J. Math. Anal., 5, 53-57 (1974) · Zbl 0291.33010
[4]	Askey, R.; Gasper, G., Certain rational functions whose power series have positive coefficients, Amer. Math. Monthly, 79, 327-341 (1972) · Zbl 0242.33023
[5]	Askey, R.; Gasper, G., Convolution structures for Laguerre polynomials, J. Anal. Math., 31, 48-68 (1977) · Zbl 0347.33006
[6]	Baryshnikov, Y.; Pemantle, R., Asymptotics of multivariate sequences, part III: quadratic points, Adv. Math., 228, 3127-3206 (2011) · Zbl 1252.05012
[7]	Berndt, B. C., Ramanujan’s Notebooks, Part V (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0886.11001
[8]	Borwein, J. M.; Borwein, P. B., A cubic counterpart of Jacobi’s identity and the AGM, Trans. Amer. Math. Soc., 323, 691-701 (1991) · Zbl 0725.33014
[10]	Franel, J., On a question of Laisant, L’Intermédiaire Math., 1, 45-47 (1894)
[11]	Gillis, J.; Reznick, B.; Zeilberger, D., On elementary methods in positivity theory, SIAM J. Math. Anal., 14, 396-398 (1983) · Zbl 0599.42500
[12]	Grace, J. H., The zeros of a polynomial, Proc. Cambridge Philos. Soc., 11, 352-357 (1902) · JFM 33.0121.04
[13]	Hörmander, L., On a theorem of Grace, Math. Scand., 2, 55-64 (1954) · Zbl 0058.25502
[14]	Ismail, M. E.H.; Tamhankar, M. V., A combinatorial approach to some positivity problems, SIAM J. Math. Anal., 10, 478-485 (1979) · Zbl 0409.05009
[15]	Kauers, M., Computer algebra and power series with positive coefficients, (Formal Power Series and Algebraic Combinatorics (2007), Nankai University: Nankai University Tianjin, China), 7 pp.; available from http://igm.univ-mlv.fr/ fpsac/FPSAC07/SITE07/PDF-Proceedings/Talks/20.pdf
[16]	Kauers, M.; Zeilberger, D., Experiments with a positivity-preserving operator, Exp. Math., 17, 341-345 (2008) · Zbl 1157.05003
[17]	Koornwinder, T., Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula, J. Lond. Math. Soc. (2), 18, 101-114 (1978) · Zbl 0386.33009
[18]	Maier, R. S., Algebraic hypergeometric transformations of modular origin, Trans. Amer. Math. Soc., 359, 3859-3885 (2007) · Zbl 1145.11034
[19]	Pemantle, R.; Wilson, M. C., Asymptotics of multivariate sequences. I. Smooth points of the singular variety, J. Combin. Theory Ser. A, 97, 129-161 (2002) · Zbl 1005.05007
[20]	Pemantle, R.; Wilson, M. C., Asymptotics of multivariate sequences. II. Multiple points of the singular variety, Combin. Probab. Comput., 13, 735-761 (2004) · Zbl 1065.05010
[21]	Pemantle, R.; Wilson, M. C., Twenty combinatorial examples of asymptotics derived from multivariate generating functions, SIAM Rev., 50, 199-272 (2008) · Zbl 1149.05003
[22]	Raichev, A.; Wilson, M. C., Asymptotics of coefficients of multivariate generating functions: improvements for smooth points, Electron. J. Combin., 15, R89 (2008), 17 pp · Zbl 1165.05309
[24]	Straub, A., Positivity of Szegö’s rational function, Adv. Appl. Math., 41, 255-264 (2008) · Zbl 1147.42011
[25]	Szegö, G., Über gewisse Potenzreihen mit lauter positiven Koeffizienten, Math. Z., 37, 674-688 (1933) · Zbl 0007.34401