Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.4

Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions

Maxie D. Schmidt
University of Washington
Department of Mathematics
Padelford Hall
Seattle, WA 98195


The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi-type continued fractions (J-fractions) that formally enumerate the typically divergent ordinary generating functions of these sequences. The rational convergents of these generalized J-fractions provide formal power series approximations to the ordinary generating functions that enumerate many specific classes of factorial-related integer product sequences. The article also provides applications to a number of specific factorial sum and product identities, new integer congruence relations satisfied by generalized factorial-related product sequences, the Stirling numbers of the first kind, and the r-order harmonic numbers, as well as new generating functions for the sequences of binomials, mp - 1, among several other notable motivating examples given as applications of the new results proved in the article.

Full version:  pdf,    dvi,    ps,    latex     Mathematica notebook    

(Concerned with sequences A000043 A000108 A000142 A000165 A000166 A000178 A000215 A000225 A000407 A000668 A000918 A000978 A000984 A001008 A001044 A001097 A001142 A001147 A001220 A001348 A001359 A001448 A002109 A002144 A002234 A002496 A002805 A002981 A002982 A003422 A005109 A005165 A005384 A006512 A006882 A007406 A007407 A007408 A007409 A007540 A007559 A007619 A007661 A007662 A007696 A008275 A008277 A008292 A008544 A008554 A009120 A009445 A010050 A019434 A022004 A022005 A023200 A023201 A023202 A023203 A024023 A024036 A024049 A027641 A027642 A032031 A033312 A034176 A046118 A046124 A046133 A047053 A061062 A066802 A077800 A078303 A080075 A085157 A085158 A087755 A088164 A094638 A100043 A100089 A100732 A104344 A105278 A123176 A130534 A157250 A166351 A184877.)

Received January 5 2016; revised versions received March 13 2016; April 7 2016; December 24 2016; December 29 2016. Published in Journal of Integer Sequences, January 8 2017.

Return to Journal of Integer Sequences home page