Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.3

Some Classes of Numbers and Derivatives

Milan Janjić
Department of Mathematics and Informatics
University of Banja Luka
Republic of Srpska, Bosnia and Herzegovina


We prove that three classes of numbers -- the non-central Stirling numbers of the first kind, generalized factorial coefficients, and Gould-Hopper numbers -- may be defined by the use of derivatives. We derive several properties of these numbers from their definitions. We also prove a result for harmonic numbers. The coefficients of Hermite and Bessel polynomials are a particular case of generalized factorial coefficients, The coefficients of the associated Laguerre polynomials are a particular case of Gould-Hopper numbers. So we obtain some properties of these polynomials. In particular, we derive an orthogonality relation for the coefficients of Hermite and Bessel polynomials.

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(Concerned with sequences A000369 A000522 A001497 A001701 A001702 A001705 A001706 A001707 A001708 A001709 A001711 A001712 A001713 A001716 A001717 A001718 A001722 A001723 A001724 A001801 A004747 A008297 A013988 A021009 A035342 A035469 A049029 A049385 A049444 A049458 A049459 A049600 A051338 A051339 A051379 A051523 A051524 A051525 A051545 A051546 A051560 A051561 A051562 A051563 A051564 A051565 A059343 A072019 A072020 A084358 A092082 A094587 A105278 A111596 A122850 A132013 A132014 A132056 A132062 A132159 A132681 A132710 A132792 A136215 A136656.)

Received August 4 2009; revised version received November 19 2009. Published in Journal of Integer Sequences, November 25 2009. Minor correction, January 29 2010.

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