A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould–Quaintance formulas

Letting B n ( x ) the n -th Bell polynomial, it is well known that B n admit specific integer coordinates in the two following bases { x i } i = 0 , … , n and { x B i ( x ) } i = 0 , … , n − 1 according, respectively, to Stirling numbers and binomial coefficients. Our aim is to prove that, for r + s = n , the sequence { x j B k ( x ) } j = 0 , … , r k = 0 , … , s is a family of bases of the Q -vectorial space formed by polynomials of Q [ X ] for which B n admits a Binomial Recurrence Coefficient.