some integral representations and properties of lah

Journal for Algebra and Number Theory Academia Volume 4, Issue 3, September 2014, Pages 77-87 © 2014 Mili Publications

SOME INTEGRAL REPRESENTATIONS AND PROPERTIES OF LAH NUMBERS BAI-NI GUO and FENG QI School of Mathematics and Informatics Henan Polytechnic University Jiaozuo City, Henan Province, 454010, China E-mail: [email protected] [email protected] College of Mathematics Inner Mongolia University for Nationalities Tongliao City, Inner Mongolia Autonomous Region 028043, China and Department of Mathematics College of Science Tianjin Polytechnic University Tianjin City, 300387, China E-mail: [email protected] [email protected] [email protected] Abstract In the paper, the authors find some integral representations and discover some properties of Lah numbers.

1. Introduction In combinatorics, Lah numbers, discovered by Ivo Lah in 1955 and usually denoted by Ln, k , count the number of ways a set of n elements can 2010 Mathematics Subject Classification: 05A10, 05A19, 05A20, 11B34, 11B37, 11B65, 11B75, 11B83, 11R33, 11Y35, 11Y55, 26A48, 26A51, 33B10, 33C15, 33C20. Keywords: integral representation; convexity; total sum, Lah number; modified Bessel function of the first kind; exponential function; absolutely convex function; absolutely convex sequence; generating function. Received August 17, 2014; Revised September 11, 2014 Accepted September 12, 2014 Academic Editor: Serkan Araci

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BAI-NI GUO and FENG QI

be partitioned into k nonempty linearly ordered queues and have an explicit formula

 n  1  n! L n, k     .  k  1  k!

(1.1)

Lah numbers Ln, k  may also be interpreted as coefficients expressing rising factorials x n in terms of falling factorials x n , where

x x  1 x  2  x  n  1, x n  

n  1, n 0

(1.2)

x x  1 x  2  x  n  1,   1

n  1, n  0.

(1.3)

1

and

x

n

Lah numbers Ln, k  may be generated by

1  x k    k!  1  x 



n

x L n, k  .  n! n 0

(1.4)

For more information on Lah numbers Ln, k , please refer to [3, p. 156]. In the theory of special functions, it is well known that the modified Bessel function of the first kind I  z  may be defined [1, p. 375, 9.6.10] by

I  z  





k 0

2k   1 z   k!   k  1  2 

(1.5)

for    and z  , where  represents the classical Euler gamma function which may be defined [1, p. 255] by

 z  



0 t

z 1 t

e dt

(1.6)

for Rz  0. In this paper, we will find some integral representations and properties of Lah numbers Ln, k .

Journal for Algebra and Number Theory Academia, Volume 4, Issue 3, September 2014

SOME INTEGRAL REPRESENTATIONS AND PROPERTIES OF … 79 2. Integral Representations of Lah Numbers We first establish integral representations of Lah numbers Ln, k , in which the exponential function e 1 x and the modified Bessel function of the first kind I1 are involved. Theorem 2.1. For 1  m  n and x  0, we have n

x



e L n, k x k   x n 0 k 1

I1 2 t t n 1 2e t x dt

(2.1)

and

L n, m 

 1 dm lim I1 2 t t n 1 2 m! x 0 0 dx m



 e  x t x    dt.   n  x 

(2.2)

Proof. In [22, Theorem 1.2], among other things, it was obtained that the function

H k z   e1 z 

k

1 1  m! z m m0

(2.3)

for k  0   and z  0 has the integral representation

H k z  

 1 F 1; k  1, k  2; t t k e  z t dt k!k  1! 0 1 2



(2.4)

for Rz   0, where p Fq a1 ,  , a p ; b1 ,  , bq ; x  stands for the generalized hyper-geometric series which may be defined by p Fq a1 ,

 , a p ; b1 ,  , bq ; x  





n 0

a1 n  a p n x n b1 n  bq n n!

(2.5)

for complex numbers ai and bi  0, 1, 2,  and for positive integers

p, q  . See also [18, Section 1.2] and [19, Lemma 2.1]. When k  0, the integral representation (2.4) becomes e1 z  1 



0

I1 2 t   zt e dt t

(2.6)

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BAI-NI GUO and FENG QI

for Rz   0. By the way, the integral representation (2.6) has been applied in [5]. Hence, for n   and x  0, we have

e1 x n   1n



0

I1 2 t t n 1 2e  x t dt.

(2.7)

In [3, Chapter III, Exercise 7, p. 158], [4], and [26, Theorem 2.2], the following explicit formula for computing the n-th derivative of the exponential function e 1 x was inductively obtained:

e 1 x n   1n e 1

x

n

1k L n, k  n1 k .  x k 1

(2.8)

By the way, the formula (2.8) have been applied in [12, 13, 14, 16, 18, 19, 22]. Combining (2.7) and (2.8) and rearranging yield

e1 x

n

 k 1

L n, k 

n

 k 1

1 x nk

L n, k 

1 x

k







0



0

I1 2 t t n 1 2e  xt dt,

I1 2 t t n 1 2 x n e  xt 1 x dt,

which may be rewritten as (2.1). Differentiating 1  m  n times on both sides of (2.1) results in n



L n, k 

k m

k! x k m  k  m!



dm I1 2 t t n 1 2 0 dx m 

 e  x t x   dt.   n  x 

Letting x  0  in the above equation leads to (2.2). The proof of Theorem 2.1 is complete.



3. Convexity of Total Sum of Lah Numbers With the help of the integral representation (2.1), we are now in a position to prove the convexity of the total sum of Lah numbers. Theorem 3.1. For n  , the total sum of Lah numbers

Journal for Algebra and Number Theory Academia, Volume 4, Issue 3, September 2014

SOME INTEGRAL REPRESENTATIONS AND PROPERTIES OF … 81 n

n 

L n, k   k 1

(3.1)

is a convex sequence. Proof. Letting x  1 in (2.1) gives

n 



0

I1 2 t t n 1 2e  1 t dt.

It is clear that the function t x for t  0 satisfies result, when t  0, the function t

x

d kt x dx

k

 t x ln t k . As a

is convex with respect to x, that is,

tx  t y  t x  y  2 , t  0, x , y  . 2 Taking in the above inequality x  n  1 and y  n  1 for n   yields

t n 1  t n 1  t n , t  0, n  . 2 Consequently, we have   1  n 1 2  1 t    I 2 t t e dt  I1 2 t t n  21 2e 1 t dt  1  2 0 0 









0

I1 2 t t n 11 2e  1 t dt,

that is, n  n  2  n 1 , n  . 2

Hence, the sequence  n is convex. The proof of Theorem 3.1 is complete.  4. Two Recoveries of the Formula (1.1) Finally, as by-product, two recoveries of the formula (1.1) for Lah numbers Ln, k  may be carried out as follows.

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4.1. First Recovery. The generating function (1.4) may be rewritten as k 1k 1  x   k!  1  x 



n

1n L n, k  x .  n! nk

(4.1)

The equation (4.1) may be reformulated as

1k 1



1



k! 1  x k

 nk

1n L n, k  x

nk

n!





n

1n  k L n  k, k  x .  n  k ! n 0

Because

1

1  x k



 1 t k 1 e  1  x t dt, k  1! 0



(4.2)

we have  1 1 t k 1e  1  x t dt  k! k  1! 0







1n L n  k, k 

n 0

xn . n  k !

Differentiating m times with respect to x on both sides of the above equation gives  1 t m  k 1e  1  x t dt  k! k  1! 0

1m 1







1n L n  k, k 

n m

n! x n m . n  m! n  k !

Taking x  0  in the above equation yields  1 m! t m  k 1e t dt  1m L m  k, k  m  k ! k! k  1! 0

1m 1



which may be rearranged as

L m  k, k   

m  k ! 1 m!

 1 t m  k 1e t dt k! k  1! 0



m  k ! 1 m  k  1! m  k !  m  k  1    . m! k! k  1! k!  k 1 

The formula (1.1) is thus recovered. 4.2. Second recovery. This recovery, which is simpler than the first one above, is supplied by the anonymous referee. Newton’s binomial series for negative exponents gives us Journal for Algebra and Number Theory Academia, Volume 4, Issue 3, September 2014

SOME INTEGRAL REPRESENTATIONS AND PROPERTIES OF … 83 

1 k

1  x 



    n 0

k  n  1 n x . k 1 

Multiplying both sides by x k , followed by a variable shift on the sum, yields

xk

1  x k









 k  n  1 n  k  n  1 n  n  1 x n  .  x  x   n! k 1  k  1 k  1  n!    n 0 nk nk







Dividing both sides by k! gives

1  x k    k!  1  x 



 n  1  n! x n ,    k  1  k! n! nk



which means the formula (1.1). 5. Remarks Remark 5.1. An infinitely differentiable function f on an interval I is called absolutely convex on I if f 2k  x   0 on I. See either [11, p. 375, Definition 3], or [17, p. 2731, Definition 4.5], or [23, p. 617, Definition 3], or [24, p. 3356, Definition 3]. A sequence  n 0 is said to be absolutely convex if its elements are non-negative and its successive differences satisfy

2kn  0

(5.1)

for n, k  0, where

kn 

k

k

1m  n  k  m .  m m0

(5.2)

Motivated by Theorem 3.1 and suggested by some results on interpolation by completely monotonic functions in, for example, [9] and [25, pp. 163-166], we conjecture that the total sum  n of Lah numbers Ln, k  is an absolutely convex sequence. Remark 5.2. In the early morning of 30 December 2013, the authors searched out the paper [4] in which the formula (2.8) was also found Journal for Algebra and Number Theory Academia, Volume 4, Issue 3, September 2014

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BAI-NI GUO and FENG QI

independently by five approaches. The motivation of the paper [4] is different from the ones of [26]. The motivations of the formula (2.8) in [26] essentially originated from the articles [6, 8, 10]. For more information, please refer to the expository and survey articles [15, 20, 21]. Remark 5.3. In an-email bearing the date March 7, 2014, Dr. Miloud Mihoubi in Algeria recommended the paper [2] to the authors. In [2, Lemma], it was stated that if

Pm, k x  

m

Lk m, n x n ,  n 1

(5.3)

then the m roots of Pm, k x  are real, distinct, and non-positive for all m  , where the associated Lah numbers Lk m, n for k  0 may be defined by

Lk m, n  

m! n!

n

n

1n  r     r  r 1

m  rk  1   m 

(5.4)

and Lk m, n  0 for n  m. Since L1 m, n  Lm, n, see [2, p. 158, Equation (4)], when k  1, the polynomial Pm, k x  becomes

Pm, 1 x  

m

L m, n x n .  n 1

(5.5)

For n  , the integer polynomial

 n x  

n

L n  1, k  1x k  k 0

(5.6)

of degree n satisfies

 n x  

Pn 1, 1 x  x

n



 k 0

L n  1, k  1 x k 

n 1

L n  1, k x k 1  0.  k 1

(5.7)

This shows a connection between n x  and Pm, k k . Remark 5.4. In [13], an explicit formula for Bell numbers Bn in terms of Stirling numbers of the second kind S n, k  together with Lah numbers

Ln, k  respectively was established as follows: For n  , Bell numbers Bn

Journal for Algebra and Number Theory Academia, Volume 4, Issue 3, September 2014

SOME INTEGRAL REPRESENTATIONS AND PROPERTIES OF … 85 may be expressed as n

Bn 

 k 1

 k



 Lk, S n, k.   1

1n  k 

(5.8)

This equation may be rewritten as n

1n  k ak S n, k   Bn ,  k 1

(5.9)

where a k is sequence A000262 in the Online Encyclopedia of Integer Sequences. Such a sequence a k has a nice combinatorial interpretation: it counts “the sets of lists, or the number of partitions of 1, 2,  , k into any number of lists, where a list means an ordered subset.” This reveals the combinatorial interpretation of the total sum  k of Lah numbers Lk,  . Another combinatorial interpretation of the sum  k of Lah numbers

Ln, k , pointed out by the referee, states that the total sum  k is clearly the number of ways to partition n elements into any number of nonempty lists. Remark 5.5. This paper is a corrected and modified version of the preprint [7]. Acknowledgments The authors appreciate anonymous referees for their careful corrections to and valuable comments on the original version of this paper. References [1]

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