Geodesic
Multiplicative functions generalized binomial coefficients, and generalized Catalan numbers
Journal of integer sequences, Tome 19 (2016) no. 1
We investigate generalized binomial coefficients of multiplicative functions. We provide a formula for these coefficients and use this formula to prove that the coefficients are always integral if the function is also a divisible function. Furthermore, we prove that multiplicative and divisible functions have integral generalized Fuss-Catalan numbers. Along the way, we include some results about specific multiplicative functions such as $gcd_{k}$ and $\phi $. We finish by connecting these results to a classical result due to Ward.
Classification : 11B65, 05A10, 11A05
Keywords: multiplicative function, divisible function, generalized binomial coefficient, Kummer's theorem, fuss-Catalan number 
@article{JIS_2016__19_1_a6,
     author = {Edgar,  Tom and Spivey,  Michael Z.},
     title = {Multiplicative functions generalized binomial coefficients, and generalized {Catalan} numbers},
     journal = {Journal of integer sequences},
     year = {2016},
     volume = {19},
     number = {1},
     zbl = {1364.11050},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a6/}
}
TY  - JOUR
AU  - Edgar,  Tom
AU  - Spivey,  Michael Z.
TI  - Multiplicative functions generalized binomial coefficients, and generalized Catalan numbers
JO  - Journal of integer sequences
PY  - 2016
VL  - 19
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a6/
LA  - en
ID  - JIS_2016__19_1_a6
ER  - 
%0 Journal Article
%A Edgar,  Tom
%A Spivey,  Michael Z.
%T Multiplicative functions generalized binomial coefficients, and generalized Catalan numbers
%J Journal of integer sequences
%D 2016
%V 19
%N 1
%U http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a6/
%G en
%F JIS_2016__19_1_a6
Edgar,  Tom; Spivey,  Michael Z. Multiplicative functions generalized binomial coefficients, and generalized Catalan numbers. Journal of integer sequences, Tome 19 (2016) no. 1. http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a6/