Geodesic
Explicit and Recursive Formulas, Integral Representations, and Properties of the Large Schr�der Numbers
Kragujevac Journal of Mathematics, Tome 41 (2017) no. 1, p. 121 .

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In the paper, the authors review and survey some new results on the large Schr�der numbers. These results were obtained by the authors and their coauthors from January 2016 and can be summarized up as follows. (a) By studying the generating function of the large Schr�der numbers, the authors and their coauthors establish several explicit formulas and a recursive formula for the large Schr�der numbers and present relations between the Schr�der numbers and central Delannoy numbers. (b) By utilizing the Cauchy integral formula in the theory of complex functions, the authors and their coauthors build several integral representations for the large Schr�der numbers and their generating function. (c) By employing integral representations for the large Schr�der numbers, the authors and their coauthors find some properties, including the convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schr�der numbers.
Classification : 11B83 05A15, 11R33, 11Y55, 26A48, 30E20, 33B99, 44A10, 44A15
Keywords: Large Schr�der number, little Schroder number, generating function, explicit formula, recursive formula, integral representation, convexity, logarithmic convexity, complete monotonicity, determinantal inequality, product inequality, central Delannoy numbers, relation 
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     author = {Feng Qi and Bai-Ni Guo},
     title = {Explicit and {Recursive} {Formulas,} {Integral} {Representations,} and {Properties} of the {Large} {Schr�der} {Numbers}},
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Feng Qi; Bai-Ni Guo. Explicit and Recursive Formulas, Integral Representations, and Properties of the Large Schr�der Numbers. Kragujevac Journal of Mathematics, Tome 41 (2017) no. 1, p. 121 . http://geodesic.mathdoc.fr/item/KJM_2017_41_1_a7/