[mathematical expression not reproducible] (5) where [g.sub.k] = 2[B.sub.k] - [2.sup.k+1] [B.sub.k] is the Genocchi number. [B.sub.n], [B.sub.n](x) are the Bernoulli number and Bernoulli polynomial, respectively.
where [g.sub.n-r] = [G.sub.n-r](0) is the Genocchi number, [n.sup.(r)], [(m + 1).sup.(r+1)] are the falling and rising factorial, respectively.
[G.sub.i-k] is the Genocchi number
and [c.sub.j] can be obtained from (23).
Mathematicians have studied different kinds of the Euler, Bernoulli, Tangent, and Genocchi numbers
and polynomials (see [1-14]).
The sequence of numbers of irreducible k-shapes [([P.sub.k](1)).sub.k[greater than or equal to]1] seemed to be the sequence of Genocchi numbers
[([G.sub.2k]).sub.k[greater than or equal to]1] = (1,1,3,17,155,2073, ...) [OEI], which may be defined by [G.sub.2k] = [Q.sub.2k-2(] 1) for all k [greater than or equal to] 2 (see [Car71, RS73]) where [Q.sub.2n](x) is the Gandhi polynomial defined by the recursion [Q.sub.2](x) = [chi square] and
Other topics include primitives of p-adic meromorphic functions, the Lipschitz condition for rational functions on ultrametric valued functions, the geometry of p-adic fractal strings, identities and congruencies for Genocchi numbers
, and ultrametric calculus in field K.
Lehmer in 1934 extended these methods to Euler numbers, Genocchi numbers
, and Lucas numbers (1934) , and calculated the 196-th Bernoulli numbers.
Sen, "Some new formulae for Genocchi numbers
and polynomials involving Bernoulli and Euler polynomials," International Journal of Mathematics and Mathematical Sciences, vol.
Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers
. Selecta Mathematical 15:105-119,2009.
Recently, many mathematicians have studied in the areas of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers
and polynomials, Stirling numbers, and central factorial numbers (see [1-17]).
Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi numbers
and polynomials (see [1-19]).
model for the Genocchi numbers
and Gandhi polynomials.