Asia Mathematika

An International Journal. ISSN: 2457-0834 (online)

Volume 2, Issue 1, April 2018, pp 17-19

On an identity of Spivey for Bell numbers

B. Man Tuladhar , H. Torres-Silva and J. López-Bonilla

1 Kathmandu University, Dhulikhel, Kavre, Nepal.
2 Escuela de Ingeniería Eléctrica y Electrónica, Universidad de Tarapacá, Arica, Casilla 6-D, Chile.
3 ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 5, 1er. Piso, Col. Lindavista CP 07738, CDMX, México


We give an elementary proof of the Spivey’s identity for the Bell numbers B(k), and we show that it implies a recurrence relation for ∑ j=0n Sn[j], thus these quantities involving the Stirling numbers of the second kind are linear combination of the B(k).


Matrix exponential function, Characteristic polynomial, Faddeev’s matrices, Laplace transform.



1. Spivey M Z, A generalized recurrence for Bell numbers, J. Integer Sequences, 2008 11, No.2, Art. 08.2.5.

2. Xu A, Extensions of Spivey’s Bell number formula, The Electr. J. of Combinatorics 2012, 19, No. 2 :1-8.

3. Gould HW, T. Glatzer T, Catalan and Bell numbers: Research bibliography of two special number sequences 1979,

4. Quaintance J, Gould HW, Combinatorial identities for Stirling numbers, World Scientific, Singapore 2016

5. López-Bonilla J, López-Vázquez R, Yaljá Montiel-Pérez J, An identity involving Bell and Stirling numbers, Prespacetime Journal 2017, 8, No. 12, 1391-1393

6. Barrera-Figueroa V, López-Bonilla, López-Vázquez R, On Stirling numbers of the second kind, Prespacetime Journal 2017, 8, No. 5, 572-574

7. Man Tuladhar B, J. López-Bonilla J, Salas-Torres O, An identity for Stirling numbers of the second kind, Kathmandu Univ. J. of Sci., Eng. & Tech. 2017, 13, No. 1, 95-97

8. Dobinski G, Archiv. der Mathematik und Physik. 1887, 61, 333-336.

9. López-Bonilla J, Yaljá Montiel-Pérez J, Vidal-Beltrán S, On the Melzak and Wilf identities, J. of Middle East and North Africa Sci. 2017, 3, No. 11, 1-3.

10. Cruz-Santiago R, López-Bonilla J, López-Vázquez R, On the series transformation formula of Boyadzhiev, Prespacetime Journal. 2017, 8, No. 12, 1367-1369.

11. Arakawa T, Ibukiyama T, Kaneko M, Bernoulli numbers and zeta functions, Springer, Japan 2014.

12. Knopf PM, The operator (x d/dx)nand its application to series, Math. Mag. 2003 76, No. 5,364-371.

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