CONVOLVING THE m-TH POWERS OF THE CONSECUTIVE INTEGERS WITH THE GENERAL FIBONACCI SEQUENCE USING CARLITZ’S WEIGHTED STIRLING POLYNOMIALS OF THE SECOND KIND

Summation rules for various types of convolutions have been the subject of much interest in The Fibonacci Quarterly. The following references are of particular relevance to this topic: Cohen and Hudson [8], Corley [9], Filipponi and Freitag [11], Gould [14], Haukkanen [15], Hsu [19], and finally Philippou and Georghiou [23]. Also consult Neuman and Schonbach’s SIAM Review article [22], where sums of convolved powers of the integers are determined with the help of Bernoulli numbers. In a separate area of research, the Stirling numbers and their various generalizations have also been the subject of sustained attention in The Fibonacci Quarterly. The reader is referred to Branson [2], Cacoullos and Papageorghiou [3], Cakić [4], Carlitz [5] and [6], Charalambides [7], El-Desouky [10], Fray [12], Hillman, Mana and McAbee [16], Howard [17] and [18], Khan and Kwong [20], Sitgreaves [24], Toscano [25], and finally Yu [26]. In [19], Hsu relates Stirling numbers of the second kind to a summation formula. In [14], Gould makes use of Stirling numbers of the second kind to reconsider the sums of convolved powers of the integers of Neuman and Schonbach [22]. In [7], Charalambides discusses some combinatorial applications of the weighted Stirling numbers introduced by Carlitz in [5] and [6]. In the present Note, the weighted Stirling numbers of the second kind introduced by Carlitz in [5] and [6] are used to formulate a convolution of the general Fibonacci sequence {Gn ≡ Aα +Bβn}+∞ n=−∞ with the sequence of the integral powers of the consecutive integers, {(a+ n)m}+∞ n=−∞. A few applications are also presented at the end of the Note. The following Theorem is established in the present Note: Theorem: “For m ≥ 0, a, b integers and for A,B, α, β real numbers, with α + β = 1, αβ = −1, the generalized convolution of the sequence of powers of the consecutive integers, {(a + n)m}+∞ n=−∞, with the general Fibonacci sequence, {Gn ≡ Aα +Bβn}+∞ n=−∞ is