1 Introduction

As is known, the Stirling numbers of the first kind are defined by

$$ (x)_{n} = \sum_{l=0}^{n} S_{1} (n,l) x^{l}\quad (n \ge0) \text{ (see [1, 2, 4, 7--25]}), $$
(1)

where \((x)_{0} = 1\), \((x)_{n} = x(x-1) \cdots(x-n+1)\) (\(n \ge1\)).

For any real number λ, the λ-analogue of \((x)_{n}\) is defined as

$$ (x)_{0, \lambda} = 1,\qquad (x)_{n, \lambda} = x(x - \lambda) (x- 2 \lambda) \cdots\bigl(x-(n-1) \lambda\bigr)\quad (n \ge1). $$
(2)

Note that \(\lim_{\lambda\rightarrow1} (x)_{n, \lambda} = (x)_{n}\) (\(n \ge0\)) (see [2, 11, 13, 16, 19]).

The λ-analogues of the Stirling numbers of the first kind are given by

$$ (x)_{n, \lambda} = \sum_{k=0}^{n} S_{1,\lambda} (n,k) x^{k} \quad( \text{see [13]}). $$
(3)

We recall that the λ-binomial coefficients are defined by the generating function

$$ (1+ \lambda t)^{ \frac{x}{\lambda}} = \sum_{l=0}^{\infty} { \binom{x}{l}}_{\lambda} t^{l} = \sum _{l=0}^{\infty} (x)_{l, \lambda} \frac{t^{l}}{l!} \quad ( \text{see [13, 16]}). $$
(4)

From (4), we note that

$$ { \binom{n}{k}}_{\lambda} = \frac{(n!)_{\lambda}}{ k!( n- k \lambda)_{n-k, \lambda}} = \frac{ (n)_{k, \lambda}}{k!}\quad (n \ge k \ge0), $$
(5)

where

$$ (n!)_{\lambda} = n (n- \lambda) (n- 2 \lambda) \cdots\bigl(n - (n-1) \lambda\bigr) = (n)_{n, \lambda} \quad ( \text{see [13]}) $$

and

$$ \sum_{m=0}^{n} { \binom{y}{ m}}_{\lambda} { \binom{x}{ n-m}}_{\lambda} = { \binom{x+y}{ n}}_{\lambda}\quad (n \ge0 ). $$
(6)

For \(r \in\mathbb{N}\), the unsigned r-Stirling numbers of the first kind are defined by

(x+r)(x+r+1)⋯(x+r+n−1)=∑k=0n[n+rk+r]rxk(see [10, 13, 16]).
(7)

In [16, 18], the r-Stirling numbers of the first kind are given by

$$ (x+r)_{n} = \sum_{k=0}^{n} S_{1}^{(r)} (n, k) x^{k} \quad (n \ge0). $$
(8)

By (7) and (8), we get

S1(−r)(n,k)=(−1)n−k[n+rk+r]r(n≥k≥0).
(9)

It is known that λ-analogues of r-Stirling numbers of the first kind are given by

$$ (x+r)_{n, \lambda} = \sum_{k=0}^{n} S_{1, \lambda}^{(r)} (n,k) x^{k} \quad ( \text{see [16]}). $$
(10)

From (3), (6), and (10), we note that

$$ S_{1, \lambda}^{(r)} (n,k) = \sum _{m=k}^{n} \binom{n}{m}S_{1,\lambda} (m,k) (r)_{n-m, \lambda}\quad (n,k \ge0). $$
(11)

If X is a discrete random variable taking values in the nonnegative integers, then the probability generating function of X is defined as follows:

$$ G(t) = E\bigl[t^{X}\bigr] = \sum _{x=0}^{\infty} p(x) t^{x} \quad ( \text{see [7, 24]}), $$
(12)

where \(p(x)\) is the probability mass function of X.

Let \(X= (X_{1}, X_{2} , \ldots, X_{k} )\) be a discrete random variable taking values in the k-dimensional nonnegative integer lattice. Then the probability generating function of X is defined as follows:

$$ \begin{aligned}[b] G(t) & = G( t_{1}, t_{2}, \ldots, t_{k} ) = E\bigl[ t_{1}^{X_{1}}, t_{2}^{X_{2}}, \ldots, t_{k}^{X_{k}} \bigr] \\ &= \sum_{x_{1}, x_{2}, \ldots, x_{k} =0}^{\infty} p(x_{1} , x_{2} , \ldots, x_{k} ) t_{1}^{x_{1}} t_{2}^{x_{2}} \cdots t_{k}^{x_{k}}, \end{aligned} $$
(13)

where \(p(x_{1}, x_{2}, \ldots, x_{k})\) is the probability mass function of X. The power series converges absolutely at least for all convex vectors \(t= (t_{1} , t_{2} , \ldots, t_{k} ) \in\mathbb{C}^{k}\) with \({\max\{ | t_{1} |, |t_{2} |, \ldots, | t_{k} | \} \le1}\).

The logarithmic random variable X with parameter \(\alpha\in (0,1)\) is a discrete random variable on \(\mathbb{N}\) with probability mass function \(p(x)\) given by

$$ P[X=n] = p(n) = - \frac{1}{\log(1- \alpha)} \cdot\frac{ \alpha^{n}}{n} \quad ( n \in\mathbb{N}). $$
(14)

Note that

$$ \sum_{n=1}^{\infty} p(n) = - \frac{1}{\log(1- \alpha)} \sum_{n=1}^{\infty} \frac{ \alpha^{n}}{n} =1 $$

and

$$ E[X] = \sum_{n=1}^{\infty} p(n) \cdot n = - \frac{1}{\log(1- \alpha)} \sum_{n=1}^{\infty} \alpha^{n} = - \frac{1}{\log(1- \alpha)} \cdot\frac{\alpha}{ 1- \alpha}. $$

In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution derived from the Maclaurin series expansion

$$ - \log( 1- \alpha) = \alpha+ \frac{ \alpha^{2}}{2} + \frac{ \alpha^{3}}{3} + \cdots. $$

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified number of failures (denoted by r) occurs. The negative binomial random variable is sometimes defined in terms of the random variable Y= the number of failures before the rth success. The probability mass function of the negative binomial random variable with parameters r and p is given by

$$ P[X=y] = p(y) = \binom{y+r-1}{y}p^{y}(1-p)^{r}. $$
(15)

In this paper, we consider λ-Stirling polynomials of the first kind and truncated λ- Stirling polynomials of the first kind rising from the λ-analogues of the falling factorial sequence and investigate some properties for these polynomials. In particular, we give some identities, recurrence relations, and explicit expressions for the λ-Stirling polynomials of the first kind and the truncated λ-Stirling polynomials of the first kind. Further, we show that both of them appear in an expression of the probability mass function of a suitable discrete random variable, constructed from λ-logarithmic and negative λ-binomial distributions.

2 λ-Stirling polynomials of the first kind

Let t be a real variable, x be a real number, and let n be a nonnegative integer. The Taylor expansion of the function \((t)_{n, \lambda}\) is given by

$$ \begin{aligned} (t)_{n,\lambda} & = \sum _{k=0}^{\infty} \frac{1}{k!} \biggl[ \frac{d^{k}}{dt^{k}} (t)_{n,\lambda} \biggr]_{t=x} (t-x)^{k} . \end{aligned} $$
(16)

Let

$$ \begin{aligned} S_{1,\lambda}^{(x)}(n,k) & = \frac{1}{k!} \biggl[ \frac{d^{k}}{dt^{k}} (t)_{n,\lambda} \biggr]_{t=x} \quad ( n, k \ge0). \end{aligned} $$
(17)

Then, by (16) and (17), we get

$$ \begin{aligned} (t)_{n,\lambda} & = \sum _{k=0}^{n} S_{1,\lambda}^{(x)}(n,k) (t-x)^{k}\quad (n \ge0). \end{aligned} $$
(18)

Here \(S_{1,\lambda}^{(x)}(n,k)\) will be called the λ-Stirling polynomials of the first kind.

It is easy to show that

$$ (t)_{n+1,\lambda} = (t-x) (t)_{n,\lambda} + (x-n\lambda) (t)_{n,\lambda}. $$
(19)

From (18), we can derive the following equation:

$$ \begin{aligned}[b] \sum_{k=0}^{n+1} S_{1, \lambda}^{(x)} (n+1,k) (t-x)^{k}& = (t)_{n+1,\lambda} = (t-x) (t)_{n,\lambda} + (x-n\lambda) (t)_{n,\lambda} \\ &= \sum_{k=0}^{n} S_{1, \lambda}^{(x)} (n,k) (t-x)^{k+1} + (x- n \lambda) \sum_{k=0}^{n} S_{1, \lambda}^{(x)} (n,k) (t-x)^{k} \\ &= \sum_{k=1}^{n+1} S_{1, \lambda}^{(x)} (n,k-1) (t-x)^{k} + (x- n \lambda) \sum_{k=0}^{n} S_{1, \lambda}^{(x)} (n,k) (t-x)^{k} \\ &= \sum_{k=0}^{n+1} \bigl( S_{1, \lambda}^{(x)} (n,k-1) + (x- n \lambda) S_{1, \lambda}^{(x)} (n,k) \bigr) (t-x)^{k}. \end{aligned} $$
(20)

Therefore, by comparing the coefficients on both sides of (20), we obtain the following theorem.

Theorem 2.1

For \(n,k \ge0\)with \(n \ge k-1\), we have

$$ S_{1, \lambda}^{(x)} (n,k-1) + (x- n \lambda) S_{1, \lambda}^{(x)} (n,k) = S_{1, \lambda}^{(x)} (n+1,k). $$

Note that

$$ S_{1, \lambda}^{(x)} (0,0) = 1,\qquad S_{1, \lambda}^{(x)} (n,0) = (x)_{n, \lambda}, \qquad S_{1, \lambda}^{(x)} (0,k) = 0 \quad (k > 0). $$

From (18), we easily note that \(S_{1, \lambda}^{(x)} (n,k) =0 \) if \(k > n\). Let us take \(t = x+1\) and \(t=x-1\) in (18). Then we have

$$ (x+1)_{n,\lambda} = \sum_{k=0}^{n} S_{1, \lambda}^{(x)} (n,k) (x+1-x)^{k} = \sum _{k=0}^{n} S_{1, \lambda}^{(x)} (n,k) $$

and

$$ (x-1)_{n,\lambda} = \sum_{k=0}^{n} S_{1, \lambda}^{(x)} (n,k) (-1)^{k}. $$

By (3), we get

$$ \begin{aligned}[b] \frac{1}{k!} \biggl[ \frac{ d^{k}}{dt^{k}} (t)_{n, \lambda} \biggr]_{t=x} & = \frac{1}{k!} \sum_{l=k}^{n} S_{1,\lambda} (n,l) (l)_{k} t^{l-k} \big|_{t=x} \\ &= \sum_{l=k}^{n} S_{1,\lambda} (n,l) \binom{l}{k} x^{l-k}. \end{aligned} $$
(21)

From (17) and (21), we obtain the following theorem.

Theorem 2.2

For \(n \ge k\), we have

$$ S_{1,\lambda}^{(x)} (n,k) = \sum_{l=k}^{n} S_{1,\lambda} (n,l) \binom{l}{k} x^{l-k}. $$

Note that \(S_{1,\lambda}^{(0)} (n,k) = S_{1, \lambda} (n, l)\).

Now, we give an explicit expression for the polynomials \(S_{1,\lambda}^{(x)} (n,k)\) and their relations with λ-Stirling numbers of the first kind. First we observe that

$$ \begin{aligned}[b] (1 + \lambda t)^{ \frac{y}{\lambda}} & = \sum _{k=0}^{\infty} (y)_{k,\lambda} \frac{t^{k}}{k!} \\ &= \sum_{k=0}^{\infty} \Biggl( \sum _{n=0}^{k} \frac{1}{n!} \biggl[ \frac{d^{n}}{dy^{n}} (y)_{k,\lambda} \biggr]_{y=x} (y-x)^{n} \Biggr) \frac{t^{k}}{k!} \\ &= \sum_{n=0}^{\infty} \Biggl( \sum _{k=n}^{\infty} S_{1,\lambda}^{(x)} (k,n) \frac{t^{k}}{k!} \Biggr) (y-x)^{n} . \end{aligned} $$
(22)

On the other hand,

$$ \begin{aligned}[b] (1 + \lambda t)^{ \frac{y}{\lambda}} & = e^{\frac{y}{\lambda} \log ( 1+ \lambda t)} \\ &= \sum_{k=0}^{\infty} \frac{1}{k!} \biggl( \frac{1}{\lambda} \log ( 1+ \lambda t) \biggr)^{k} e^{\frac{x}{\lambda} \log( 1+ \lambda t)} (y-x)^{k} \\ &= \sum_{k=0}^{\infty} \frac{1}{k!} \biggl( \frac{1}{\lambda} \log( 1+ \lambda t) \biggr)^{k} ( 1+ \lambda t)^{\frac{x}{\lambda}} (y-x)^{k}. \end{aligned} $$
(23)

From (22) and (23), we obtain the generating function for \(S_{1,\lambda}^{(x)}(n,k)\) given by

$$ \frac{1}{k!} \biggl( \frac{\log( 1+ \lambda t) }{\lambda} \biggr)^{k} ( 1+ \lambda t )^{ \frac{x}{\lambda}} = \sum _{n=k}^{\infty}S_{1,\lambda}^{(x)}(n,k) \frac{t^{n}}{n!}, $$
(24)

where k is a nonnegative integer.

Indeed, we note that

$$ \begin{aligned}[b] \frac{1}{k!} \biggl( \frac{\log( 1+ \lambda t) }{\lambda} \biggr)^{k} ( 1+ \lambda t )^{ \frac{x}{\lambda}} &= \sum _{m=0}^{\infty} (x)_{m, \lambda} \frac{t^{m}}{m!} \Biggl( \frac{1}{k!} \sum_{l=k}^{\infty} \biggl( \sum_{l_{1}+l_{2} + \cdots+l_{k} =l} \frac{ (- \lambda)^{l-k} }{l_{1} l_{2} \cdots l_{k} } \biggr) t^{l} \Biggr) \\ &= \sum_{n=k}^{\infty} \Biggl( \frac{1}{k!} \sum_{l=k}^{n} (- \lambda)^{l-k} l! \binom{n}{l} (x)_{n-l,\lambda} \sum _{l_{1} +l_{2} + \cdots+ l_{k} = l} \frac{1}{l_{1} l_{2} \cdots l_{k} } \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(25)

Therefore, by (24) and (25), we obtain the following theorem.

Theorem 2.3

For \(n \ge k\), we have

$$ S_{1,\lambda}^{(x)}(n,k) = \frac{1}{k!} \sum _{l=k}^{n} (- \lambda)^{l-k} l! \binom{n}{l} (x)_{n-l,\lambda} \sum_{l_{1} +l_{2} + \cdots+ l_{k} = l} \frac{1}{l_{1} l_{2} \cdots l_{k} }, $$

where the inner sum runs over all positive integers \(l_{1},l_{2},\ldots ,l_{k}\)with \(l_{1}+l_{2}+\cdots+l_{k}=l\).

It is known that

$$ \frac{1}{k!} \biggl( \frac{1}{\lambda} \log( 1 + \lambda t) \biggr)^{k} = \sum_{n=k}^{\infty} S_{1,\lambda}(n,k) \frac{t^{n}}{n!} \quad (\text{see [13, 16]}). $$
(26)

From (24) and (26), we have

$$ \begin{aligned}[b] \sum_{n=k}^{\infty} S_{1,\lambda}^{(x)}(n,k)\frac{t^{n}}{n!} & = \sum _{l=0}^{\infty} (x)_{l,\lambda} \frac{t^{l}}{l!} \sum _{m=k}^{\infty} S_{1,\lambda} (m,k) \frac{t^{m}}{m!} \\ &= \sum_{n=k}^{\infty} \Biggl( \sum _{l=0}^{n-k} (x)_{l,\lambda} \binom{n}{l} S_{1,\lambda} (n-l,k) \Biggr) \frac{t^{n}}{n!} \\ &= \sum_{n=k}^{\infty} \Biggl( \sum _{l=k}^{n} (x)_{n-l, \lambda} \binom{n}{l} S_{1,\lambda} (l,k) \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(27)

Comparing the coefficients on both sides of (27), we obtain the following theorem.

Theorem 2.4

Letn, kbe nonnegative integers. Then we have

$$ S_{1,\lambda}^{(x)}(n,k) = \textstyle\begin{cases} \sum_{l=k}^{n} \binom{n}{l} (x)_{n-l, \lambda} S_{1,\lambda} (l,k), & \textit{if } k \le n,\\ 0 ,& \textit{if } k>n. \end{cases} $$

Corollary 2.5

Letn, kbe nonnegative integers. Then we have

$$ S_{1,\lambda}(n,k) = \textstyle\begin{cases} \sum_{l=k}^{n} \binom{n}{l} (x)_{n-l, \lambda} S_{1,\lambda}^{(x)} (l,k), & \textit{if } k \le n,\\ 0 ,& \textit{if } k>n. \end{cases} $$

We now consider the r-truncated λ-Stirling numbers of the first kind.

For \(x \in\mathbb{R}\) and \(r \in\mathbb{N}\), the r-truncatedλ-Stirling polynomials of the first kind are defined by

$$ (1+ \lambda t)^{ \frac{x}{\lambda}} \frac{1}{k!} \Biggl( \frac{ \log ( 1+ \lambda t )}{\lambda} - \sum_{j=1}^{r-1} (-\lambda)^{j-1} \frac{t^{j}}{j} \Biggr)^{k} = \sum _{n=rk}^{\infty} S_{1,\lambda}^{(x)} (n, k |r) \frac{t^{n}}{n!}. $$
(28)

Remark 2.6

The definition of λ-Stirling polynomials of the first kind and that of r-truncated λ-Stirling polynomials of the first kind are similar to the non-central Stirling numbers of the first kind and the generalized non-central Stirling numbers of the first kind, respectively (see [20]). In fact, one replaces α by x and \((x)_{n}\) by \((x)_{n,\lambda}\), as in [3] and [6], or replaces α by x with setting \(\alpha_{i}=i \lambda\), \(i=0,1, \ldots,n-1\), as in (1.8) of [5].

From (28), we have

$$ \begin{aligned} [b]&(1+ \lambda t)^{ \frac{x}{\lambda}} \frac{1}{k!} \Biggl( \frac{ \log( 1+ \lambda t )}{\lambda} - \sum_{j=1}^{r-1} (-\lambda)^{j-1} \frac{t^{j}}{j} \Biggr)^{k} \\ &\quad=\sum_{m=0}^{\infty} (x)_{m, \lambda} \frac{t^{m}}{m!} \frac{1}{k!} \sum_{l=rk}^{\infty} \biggl( \sum_{l_{1}+l_{2} + \cdots+l_{k} =l} \frac{ (- \lambda)^{l-k} }{l_{1} l_{2} \cdots l_{k} } \biggr) t^{l} \\ &\quad= \sum_{n= kr}^{\infty} \Biggl( \frac{n!}{k!} \sum_{l=rk}^{n} \frac{(x)_{n-l,\lambda}}{(n-l)!} \sum_{l_{1} + \cdots+ l_{k} = l} \frac{(-\lambda)^{l-k}}{l_{1} l_{2} \cdots l_{k}} \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(29)

Therefore, by (28) and (29), we obtain the following theorem.

Theorem 2.7

For \(n \ge rk\), we have

$$ S_{1,\lambda}^{(x)} (n, k |r) =\frac{n!}{k!} \sum _{l =rk}^{n} \frac{(x)_{n-l,\lambda}}{(n-l)!} \sum _{l_{1} + \cdots+ l_{k} = l} \frac{(-\lambda)^{l-k}}{l_{1} l_{2} \cdots l_{k}}. $$

In particular, if \(n < kr\), we have

$$ S_{1,\lambda}^{(x)} (n, k |r) =0. $$

When \(x=0\), \(S_{1,\lambda}^{(0)} (n, k |r) =S_{1,\lambda} (n, k |r)\) are called the r-truncatedλ-Stirling numbers of the first kind.

It is not difficult to show that

$$ S_{1,\lambda}^{(x)} (n, k |r) = \sum_{l=kr}^{n} \binom{n}{l} (x)_{n-l, \lambda} S_{1,\lambda} (l, k |r) \quad\text{if } n \ge kr $$

and

$$ S_{1,\lambda}^{(x)} (n, k |r) = 0 \quad \text{if } n < kr. $$

Let

$$ y_{k,\lambda}^{(r)} (t |x ) = (1 + \lambda t)^{\frac{x}{\lambda}} \frac{1}{k!} \Biggl[ \frac{1}{\lambda} \log1+ \lambda t) - \sum_{j=1}^{r-1} \frac{(-\lambda)^{j-1}}{j} t^{j} \Biggr]^{k}. $$
(30)

From (30), we can derive the following differential equation:

$$ (1+ \lambda t) \frac{d}{dt} y_{k,\lambda}^{(r)} (t |x ) = x y_{k,\lambda}^{(r)} (t |x ) + (-1)^{r-1} t^{r-1} \lambda^{r-1} y_{k-1,\lambda}^{(r)} (t |x ). $$
(31)

By (28) and (31), we get

$$ \begin{aligned}[b] &(1+ \lambda t) \frac{d}{dt} y_{k,\lambda}^{(r)} (t |x )\\&\quad = x y_{k,\lambda}^{(r)} (t |x ) + (-1)^{r-1} t^{r-1} \lambda^{r-1} y_{k-1,\lambda}^{(r)} (t |x ) \\ &\quad= x \sum_{n=rk}^{\infty} S_{1,\lambda}^{(x)} (n,k |r)\frac{t^{n}}{n!}+ (-1)^{r-1} t^{r-1} \lambda^{r-1} \sum_{n=r(k-1)}^{\infty} S_{1,\lambda}^{(x)} (n,k-1 |r)\frac{t^{n}}{n!} \\ &\quad= x \sum_{n=rk}^{\infty} S_{1,\lambda}^{(x)} (n,k |r)\frac{t^{n}}{n!}+ (-1)^{r-1} \lambda^{r-1} \sum _{n=rk-1}^{\infty} S_{1,\lambda}^{(x)} (n-r+1,k-1 |r)\frac{t^{n}}{(n-r+1)!} \\ &\quad= \sum_{n=rk-1}^{\infty} \biggl( x S_{1,\lambda}^{(x)} (n,k |r)+ (-1)^{r-1} \lambda^{r-1} (r-1)! \binom{n}{r-1} S_{1,\lambda}^{(x)} (n-r+1,k-1 |r) \biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(32)

On the other hand,

$$ \begin{aligned}[b] (1+ \lambda t)\frac{d}{dt} y_{k,\lambda}^{(r)} (t |x ) & = \sum_{n= rk}^{\infty} S_{1,\lambda}^{(x)} (n,k |r) \frac{t^{n-1}}{(n-1)!} (1+ \lambda t) \\ & = \sum_{n= rk-1}^{\infty} S_{1,\lambda}^{(x)} (n+1,k |r) \frac{t^{n}}{n!} + \sum_{n= rk}^{\infty}n \lambda S_{1,\lambda}^{(x)} (n,k |r) \frac{t^{n}}{n!} \\ & = \sum_{n=rk-1}^{\infty} \bigl( S_{1,\lambda}^{(x)} (n+1,k |r) + n \lambda S_{1,\lambda}^{(x)} (n,k |r) \bigr)\frac{t^{n}}{n!}. \end{aligned} $$
(33)

Therefore, by (32) and (33), we obtain the following theorem.

Theorem 2.8

Letn, kbe nonnegative integers, and letrbe a positive integer. Then we have

$$\begin{aligned} &\lambda^{r-1} (r-1)! \binom{-n+r-2}{r-1} S_{1,\lambda}^{(x)} (n-r+1,k-1 |r) \\&\quad= S_{1,\lambda}^{(x)} (n+1,k |r) + (n \lambda-x) S_{1,\lambda}^{(x)} (n,k |r)\quad (n \ge kr-1).\end{aligned} $$

It is easy to show that

$$ \begin{aligned}[b] y_{k, \lambda}^{(r+1)}(t|x) &= ( 1+ \lambda t)^{ \frac{x}{\lambda}} \frac{1}{k!} \Biggl[ \frac{1}{\lambda} \log( 1+ \lambda t) - \sum_{j=1}^{r} (- \lambda)^{j-1} \frac{t^{j}}{j} \Biggr]^{k} \\ &= ( 1+ \lambda t)^{ \frac{x}{\lambda}} \frac{1}{k!} \Biggl[ \frac{1}{\lambda} \log( 1+ \lambda t) - \sum_{j=1}^{r-1} (- \lambda)^{j-1} \frac{t^{j}}{j} + (-1)^{r} \lambda ^{r-1} \frac{t^{r}}{r} \Biggr]^{k} \\ &= ( 1+ \lambda t)^{ \frac{x}{\lambda}} \frac{1}{k!} \sum _{l=0}^{k} \binom{k}{l} \Biggl( \frac{1}{\lambda} \log( 1+ \lambda t) - \sum_{j=1}^{r-1} (-\lambda)^{j-1} \frac{t^{j}}{j} \Biggr)^{k-l} (-1)^{rl} \frac{\lambda^{rl-l}}{r^{l}} {t^{rl}} \\ &= \sum_{l=0}^{k} \frac{(-1)^{rl}\lambda^{rl-l} }{l! r^{l}} t^{rl} \frac{1}{(k-l)!} ( 1+ \lambda t)^{\frac{x}{\lambda}} \Biggl( \frac{ \log( 1+ \lambda t)}{ \lambda} - \sum_{j=1}^{r-1} (-1)^{j-1} \frac{t^{j}}{j} \Biggr)^{k-l} \\ &= \sum_{l=0}^{k} \frac{(-1)^{rl}\lambda^{rl-l} }{l! r^{l}} t^{rl} y_{k-l, \lambda}^{(r)}(t|x). \end{aligned} $$
(34)

By (34), we get

$$ \begin{aligned}[b] \sum_{n=kr}^{\infty} S_{1,\lambda}^{(x)}( n,k|r+1) \frac{t^{n}}{n!} & = \sum _{n=k(r+1)}^{\infty} S_{1,\lambda}^{(x)}( n,k|r+1) \frac{t^{n}}{n!} \\ &= \sum_{l=0}^{k} \frac{(-1)^{rl}\lambda^{rl-l} }{l! r^{l}} t^{rl} \sum_{n=(k-l)r}^{\infty} S_{1,\lambda}^{(x)}( n,k-l|r+1) \frac{t^{n}}{n!} \\ & = \sum_{n=kr}^{\infty} \Biggl( \sum _{l=0}^{k} \frac {(-1)^{rl}\lambda^{rl-l} }{l! r^{l}} S_{1,\lambda}^{(x)}( n-lr,k-l|r) (n)_{lr} \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(35)

Comparing the coefficients on both sides of (35), we have

$$ S_{1,\lambda}^{(x)}( n,k|r+1) = \sum _{l=0}^{k} \frac{(-1)^{rl}\lambda^{rl-l} }{l! r^{l}} S_{1,\lambda}^{(x)}( n-lr,k-l|r) (n)_{lr}, $$
(36)

where \(n \ge kr \).

For \(\lambda\in(0,1)\), X is a random variable with the λ-logarithmic distribution with parameter \(\alpha\in(0,1)\) if the probability mass function of X is given by

$$ P_{\lambda}[ X = k ] = P_{\lambda}(k) = - \frac{\lambda}{ \log(1 - \alpha\lambda)} \cdot\frac{\alpha^{k} \lambda^{k-1}}{k}, $$
(37)

where k is a positive integer.

We easily see that

$$ \sum_{k=1}^{\infty} P_{\lambda}(k) = 1,\qquad E[X] = -\frac{1}{ \log(1 - \alpha\lambda)} \cdot\frac{ \alpha\lambda}{ 1- \alpha \lambda}. $$

Y is the random variable with negative λ-binomial distribution with parameters r, α if the probability mass function of Y is given by

$$ P_{\lambda} [Y=k] = P_{\lambda} (k) = \binom{ \frac{r}{ \lambda} + k -1}{k} ( \lambda \alpha)^{k} (1- \lambda\alpha)^{\frac{r}{\lambda}}, $$

where r, k, α are respectively the number of failures, the number of successes, and the probability of successes.

Let \(X_{1}, X_{2}, \ldots, X_{k}\) be independent random variables with λ-logarithmic distribution with parameter α, and let Y be the random variable with negative λ-binomial distribution with parameters r and α. If Y is independent of \(X= X_{1} + \cdots+ X_{k}\), then we have

$$ \begin{aligned}[b] E\bigl[ t^{X+Y} \bigr] & = E \bigl[t^{X}\bigr] E\bigl[t^{Y}\bigr] = \Biggl( \prod_{j=1}^{k} E \bigl[t^{X_{j}}\bigr] \Biggr) \cdot E\bigl[t^{Y}\bigr]. \end{aligned} $$
(38)

Now, we observe that

$$ E\bigl[t^{X_{j}}\bigr] = \sum _{x=1}^{\infty} P_{\lambda} [X_{j} = x] t^{x} = \frac{1}{ \log(1- \alpha\lambda)} \cdot\log(1- \alpha\lambda t) $$
(39)

and

$$ \begin{aligned}[b] E\bigl[ t^{Y}\bigr]& = \sum _{y=0}^{\infty} P_{\lambda} [Y= y] t^{y} = \sum_{y=0}^{\infty} \binom{\frac{r}{\lambda} + y -1 }{ y} (\lambda \alpha)^{y} ( 1- \lambda \alpha)^{\frac{r}{\lambda}}t^{y} \\ &= ( 1- \lambda\alpha)^{\frac{r}{\lambda}} ( 1- \lambda\alpha t )^{-\frac{r}{\lambda}}. \end{aligned} $$
(40)

From (38), (39), and (40), we have

$$ \begin{aligned}[b] E\bigl[t^{X+Y}\bigr] &= \Biggl( \prod_{j=1}^{k} E\bigl[t^{X_{j}} \bigr] \Biggr) \cdot E\bigl[t^{Y}\bigr] \\ & = \biggl(\frac{1}{\log(1- \alpha\lambda)} \biggr)^{k} \bigl( \log( 1- \alpha \lambda t) \bigr)^{k} ( 1- \alpha\lambda)^{ \frac{r}{ \lambda}} ( 1- \alpha \lambda t )^{- \frac{r}{ \lambda}} \\ &= k! \biggl(\frac{\lambda}{\log(1- \alpha\lambda)} \biggr)^{k} ( 1- \alpha \lambda)^{\frac{r}{ \lambda}} \frac{1}{k!} \biggl( \frac{\log(1- \alpha\lambda t)}{\lambda} \biggr)^{k} ( 1- \alpha\lambda t)^{-\frac{r}{\lambda}} \\ &=k! \biggl(\frac{\lambda}{\log(1- \alpha\lambda)} \biggr)^{k} ( 1- \alpha \lambda)^{\frac{r}{ \lambda}} \sum_{n=k}^{\infty} S_{1,\lambda}^{(-r)} (n,k) (-\alpha)^{n} \frac{t^{n}}{n!}. \end{aligned} $$
(41)

On the other hand,

$$ E\bigl[t^{X+Y}\bigr] = \sum _{n=k}^{\infty} P_{\lambda} [X+Y = n ] t^{n}. $$
(42)

Therefore, by (41) and (42), we obtain the following theorem.

Theorem 2.9

Let \(X_{1}, X_{2}, \ldots, X_{k}\)be independent random variables withλ-logarithmic distribution with parameterα, and letYbe the random variable with negativeλ-binomial distribution with parametersrandα. IfYis independent of \(X= X_{1} + X_{2} + \cdots+ X_{k}\), then the probability mass function of \(X+Y\)is given by

$$ P_{\lambda} [X+Y = n ] = k! \biggl( \frac{ \lambda}{ \log(1- \alpha \lambda)} \biggr)^{k} ( 1- \alpha\lambda)^{\frac{r}{x}} \frac{(-\alpha)^{n}}{n!} S_{1,\lambda}^{(-r)} (n,k) $$

for \(n \ge k\).

For \(r \in\mathbb{N}\), X is the random variable with r-truncated λ-logarithmic distribution with parameter α if the probability mass function of X is given by

$$ \begin{aligned} P_{\lambda}[X=x] &= P_{\lambda}(x) = \frac{\lambda}{ - \log(1- \alpha \lambda)- \sum_{i=1}^{r-1} \frac{ \lambda^{i} \alpha^{i}}{i} } \cdot \frac{ \alpha^{x} \lambda^{x-1}}{x} \\ & = C_{\lambda} ( \alpha, r) \frac{ \alpha^{x} \lambda^{x-1}}{x} \quad (x = r, r+1 , \ldots), \end{aligned} $$

where

$$ C_{\lambda} ( \alpha, r) = \frac{\lambda}{ - \log(1- \alpha \lambda)- \sum_{i=1}^{r-1} \frac{ \lambda^{i} \alpha^{i}}{i} }. $$

Note that \(\sum_{x=r}^{\infty} P_{\lambda}[X=x] =1\).

Let \(X_{1}, X_{2}, \ldots, X_{k}\) be independent random variables with the r-truncated λ-logarithmic distribution with parameter p, and let Y be the random variable with negative λ-binomial distribution with parameters α, p. If Y is independent of \(X= X_{1} + X_{2} + \cdots+ X_{k}\), then we have

$$ E\bigl[t^{X+Y}\bigr] = E\bigl[t^{X}\bigr] E \bigl[t^{Y}\bigr] = \Biggl( \prod_{j=1}^{k} E\bigl[t^{X_{j}}\bigr] \Biggr) E\bigl[ t^{Y} \bigr]. $$
(43)

Now, we observe that

$$\begin{aligned} E\bigl[t^{X_{j}}\bigr] & = \sum _{x=r}^{\infty} P_{\lambda} [X_{j} = x] t^{x} \\ & = C_{\lambda} (p, r) \sum_{x=r}^{\infty} \frac{\lambda^{x-1} p^{x}}{x} t^{x} \\ & = C_{\lambda} (p, r) \Biggl( \sum_{x=1}^{\infty} \frac{\lambda ^{x-1} p^{x}}{x} t^{x} - \sum_{x=1}^{r-1} \frac{\lambda^{x-1} p^{x}}{x} t^{x} \Biggr) \\ &= C_{\lambda} (p, r) \Biggl( -\frac{1}{\lambda} \log( 1- \lambda p t ) - \sum_{x=1}^{r-1} \frac{ \lambda^{x-1}p^{x}}{x}t^{x} \Biggr), \end{aligned}$$
(44)

and

$$ \begin{aligned}[b] E\bigl[t^{Y}\bigr] & = \sum _{y=0}^{\infty} \binom{ \frac{\alpha}{\lambda} + y - 1}{ y} ( \lambda p)^{y} ( 1- \lambda p)^{\frac{\alpha}{\lambda}} t^{y} \\ &= (1- \lambda p t )^{-\frac{\alpha}{\lambda}} (1- \lambda p )^{\frac{\alpha}{\lambda}}. \end{aligned} $$
(45)

From (43), (44), and (45), we have

$$ \begin{aligned}[b] E\bigl[t^{X+Y}\bigr] & = k! C_{\lambda}(p, r)^{k}(-1)^{k} \frac{1}{k!} \Biggl( \frac{\log(1- \lambda p t)}{\lambda} + \sum_{x=1}^{r-1} \frac{\lambda^{x-1}}{x} p^{x} t^{x} \Biggr)^{k} (1- \lambda p t )^{-\frac{\alpha}{\lambda}} (1- \lambda p )^{\frac{\alpha}{\lambda}}\hspace{-12pt} \\ &=k! C_{\lambda} (p, r)^{k}(1- \lambda p )^{\frac{\alpha}{\lambda}} (-1)^{k} \sum_{n=rk}^{\infty} S_{1, \lambda}^{(- \alpha)} (n,k |r) \frac{(-p)^{n}}{n!}t^{n}. \end{aligned} $$
(46)

On the other hand,

$$ E\bigl[t^{X+Y}\bigr] = \sum _{n= rk}^{\infty} P_{\lambda}[X+Y = n] t^{n}. $$
(47)

Therefore, by (46) and (47), we obtain the following theorem.

Theorem 2.10

For \(r \in\mathbb{N}\), let \(X_{1},X_{2}, \ldots, X_{k}\)be independent random variables with ther-truncatedλ-logarithmic distribution with parameterp, and letYbe the random variable with negativeλ-binomial distribution with parametersα, p. IfYis independent of \(X=X_{1} +X_{2} + \cdots+ X_{k}\), then the probability mass function of \(X+Y\)is given by

$$ P_{\lambda}[X+Y =n] = k! C_{\lambda} (p, r)^{k}(1- \lambda p )^{\frac{\alpha}{\lambda}} (-1)^{n-k} S_{1, \lambda}^{(- \alpha)} (n,k |r) \frac{p^{n}}{n!} \quad (n \ge kr), $$

where

$$ C_{\lambda} ( \alpha, r) = \frac{\lambda}{ - \log(1- \alpha \lambda)- \sum_{i=1}^{r-1} \frac{ \lambda^{i} \alpha^{i}}{i} }. $$

3 Conclusion

Stirling numbers of the first kind appear frequently in combinatorics and number theory. Recently, λ-analogues of Stirling numbers of the first kind were studied in [10].

In this paper, we introduced λ-Stirling polynomials of the first kind which appear as the coefficients in the Taylor expansion of λ-falling factorial sequence and reduce to the Stirling numbers of the first kind when \(x=0\) and \(\lambda=1\). We obtained recurrence relations, explicit expressions, some identities, and connections with other special polynomials for these polynomials. We showed that they appear in an expression of the probability mass function of a suitable discrete random variable, constructed from λ-logarithmic and negative λ-binomial distributions. Thereby we demonstrated that these polynomials are not out of nowhere but arise naturally.

We also considered r-truncated λ-Stirling polynomials of the first kind whose generating function is obtained from that of the λ-Stirling polynomials of the first kind by truncating first \(r-1\) terms in the Taylor expansion of the logarithmic function. We derived several basic properties about these polynomials just in the case of λ-Stirling polynomials of the first kind. Then we showed that they also appear in an expression of the probability mass function of a suitable discrete random variable, constructed from r-truncated λ-logarithmic and negative λ-binomial distributions. Once again, this demonstrates that r-truncated λ-Stirling polynomials of the first kind arise naturally.

As one of our future projects, we would like to continue to find many applications of λ-Stirling polynomials of the first kind and r-truncated λ-Stirling polynomials of the first kind in mathematics, sciences, and engineering.