Asymptotic properties of the Delannoy numbers and similar arrays

The Delannoy numbers were introduced and studied by Henri-Auguste Delannoy (1833–1915). He investigated the possible moves on a chessboard: the numbers under consideration appear when one studies “la marche de la Reine,” i.e., how the queen moves (the binomial coefficients appear similarly for the moves of the rook). The asymptotic behavior of the array of Delannoy numbers is studied. The regularized upper and lower radial indicators of the array are determined, proved to coincide and to be concave. We also describe the radial indicator as an infimum of linear functions, which amounts to determining its Fenchel transform. Since the methods developed for this study apply to more general convolution equations, we prove results also for these equations.