Suppose that and consider the normalized Bessel function of the first kind, defined by (1.5).
holds between the polynomials E n [ m − 1, α ] ( x ; λ ) and the Stirling numbers S ( n, k ) of the second kind defined by [[12], p.58, Eq. (15)].
S 2 ( n − j, m ) ) x j, where S 2 ( l, m ) are the Stirling numbers of the second kind, defined by ( e t − 1 ) m = m !
When x = 0, c ˆ n ( k ) = c ˆ n ( k ) ( 0 ) are called the poly-Cauchy numbers of the second kind, defined by ∑ n = 0 ∞ c ˆ n ( k ) t n n !
where ϕ(x y z) is the confluent hypergeometric function of the second kind, defined by the integral (9.211.4), [25] begin{aligned} phi (x y z)=frac{1}{Gamma(x)}int_{0}^{+infty}exp -zt t^{+infty}exp -zt t}dt.
Here, (S_{2}(n,m)) is the Stirling number of the second kind defined by the following generating series: sum_{n=m}^{infty}S_{2}(n,m) frac{t^{n}}{n!}=frac {(e^{t}-1)^{m}}{m!} quadtextit{cf.
We begin by recalling the Stirling numbers s ( n, k ) of the first kind defined by the generating functions (see, e.g., [[20], Section 1.5]; see also [[21], Section 1.6]) z ( z − 1 ) ⋯ ( z − n + 1 ) = ∑ k = 0 n s ( n, k ) z k (2.1).
Note that (lim_{lambdarightarrow0}K_{n} (xmidlambda )=b_{n} (x )), where (b_{n} (x )) are the Bernoulli polynomials of the second kind defined by the generating function biggl(frac{t}{log (1+t )} biggr) (1+t )^{x}=sum _{n=0}^{infty}b_{n} (x ) frac{t^{n}}{n!} quad (mbox{see [7, 13]} ).
where the function Ψ(x y z) is the confluent hypergeometric function of the second kind, defined by the integral Psi left({x y z} right) = frac{1}{{Gamma left(x right)}}intlimits_{0}^{infty} {e^{- zt} t^{x - 1} left({1 + t} right)^{y - x - 1} dt}.
In [23], Kim et al. introduced using p-adic integral techniques the idea that the Changee numbers are closely related to the Euler numbers as follows: E_{m}=sum_{n=0}^{m}Ch_{n}S_{2} ( n,m ), where (S_{2} ( n,m ) ) is the Stirling number of the second kind defined by the following generating series: sum_{n=m}^{infty}S_{2} ( n,m ) frac{t^{n}}{n!}=frac{ ( e^{t}-1 ) ^{m}}{m!}quadtextit{cf.
