Let $ R $ to be a commutative ring, unital. A map $ f: R times R to R $ is said to be a $ R $–*Map of Pascal* if for all $ r_1, r_2 in R $, the relationship

$$ f (r_1-1_R, r_2-1_R) + f (r_1-1_R, r_2) = f (r_1, r_2) $$

is satisfied. Example of Pascal cards include the trivial $ R $-Pascal map $ f (r_1, r_2) = 0_R $ for everyone $ r_1, r_2 in R $ and the $ mathbb {Z} $-Pascal map defined by

$$ f (n, k) = binom {n} {k}, $$

for $ 0 leq k leq n $ and $ 0 other. (Note that $ 1_R $ is not a $ R $-Pascal map unless the ring $ R $ has characteristic 1).

Let $ mathrm {Bin} (R) $ denotes the collection of all $ R $-Pascal maps; then $ mathrm {Bin} (R) $ is an abelian subgroup of $ R $ under $ + $ and $ mathrm {Bin} (R) $ is a $ R $-module under multiplication in $ R $ (in fact, we can leave $ mathrm {Bin} (R) $ be a $ mathbb {Z} $-module if we define the action as being

$$ n cdot f = start {case} onbrace {f + f cdots + f} ^ {n- text {time}}, ; & n> 0 \ 0_R, & n = 0 \ onbrace {-f – f cdots – f} ^ {(- n) – text {times}}, & n <0 end {cases} $ $

where the action is the addition / subtraction of a given term $ n $-time). In addition, $ mathrm {Bin} (R) BigProp Mathrm {Bin} (S) $ is a $ R bigoplus S $-module (by extension, a $ mathbb {Z} bigoplus mathbb {Z} $-module).

$ textbf {Question:} $ What can we say more about $ mathrm {Bin} (R) $? Is there a relationship between $ mathrm {Bin} (R) BigProp Mathrm {Bin} (S) $ and $ mathrm {Bin} (R bigoplus S) $? What can be said of a closed form of $$ sum_ {r_1, r_2 in R} f (r_1, r_2), $$

and, in fact, is there an appropriate generalization of the binomial theorem under these $ R $Cards -Pascal?$ textbf {Motivation:} $ I came across these maps after attempting to answer questions about the properties of Bhargava's factorial, including when there was an asymptotic formula of Stirling for the factorial and the question of the existence of a bounded variation function on $ (0, infty) $ whose partial moments generate the factorial of Bhargava; The answers to these questions can lead to a generalization of a zeta function of Riemann using the factorial of Bhargava via an integral representation. $ R $-Pascal maps have been developed to try to "regulate" the Bhargava factorial by determining when its binomial equivalent satisfies Pascal's identity under the addition of an ordinary integer and when a "special" addition "must be done in order for Pascal's identity to be satisfied. .