An Asymptotic Formula for the Second Kind r-Whitney Numbers with Real Parameters

An Asymptotic Formula for the Second Kind r-Whitney Numbers

with Real Parameters

1_{Cristina B. Corcino and} 2_{Raylee J. Gasparin} 1Mathematics Department, Cebu Normal University 2Department of Mathematics, Mindanao State University

Date Submitted: September 30, 2015 Originality: 85%

Date Revised: December 23, 2015 Plagiarism Detection: Passed ABSTRACT

An asymptotic formula for the r-Whitney numbers of the second kind with real parameters will be derived using Hankel’s loop integral and a modified saddle-point method.

Mathematics Subject Classification (2010). 05A10, 11B73.

Keywords: Asymptotic approximation, saddle-point method, Stirling numbers, r-Stirling numbers, Whitney numbers

INTRODUCTION

The 𝑟 -Whitney numbers are generalization of Whitney numbers (see (Benoumhani, 1996, 1997, 1999), the classical Stirling numbers and other Stirling-type numbers (see (Broder, 1984) and (Comtet, 1974)). These numbers have been introduced by Mezo (2010) in two different kinds to obtain new formulas for Bernoulli polynomials. The 𝑟 -Whitney numbers of the second kind, denoted by 𝑊𝛽,𝑟(𝑛, 𝑘), were defined in (Mezo, 2010) by means of the following linear transformation

(𝛽𝑥 + 𝑟)𝑛 = ∑𝑛𝑘=0𝛽𝑘𝑊𝛽,𝑟(𝑛, 𝑘)(𝑥)𝑘,

which coincide with the (𝑟, 𝛽)–Stirling numbers in (Corcino et al, 2011) and those numbers considered by Rucinski and Voigt (1990). These numbers satisfy the following exponential generating function

∑𝑛≥0𝑊𝛽,𝑟(𝑛, 𝑚) 𝑧𝑛

𝑛! = 1 𝛽𝑚_𝑚!𝑒

𝑟𝑧_(𝑒𝛽𝑧_{− 1)}𝑚_,

which is exactly the exponential generating function of (𝑟, 𝛽)–Stirling numbers obtained in (Corcino, 1999). More properties of these numbers can be found in (Cheon and Jung, 2012), (Corcino-Corcino, 2011), (Corcino and Aldema, 2002), (Corcino et al, 2011), (Mezo, 2010). Recently, Merca (2014) established a new connection between 𝑟-Whitney numbers and Bernoulli polynomials by expressing a finite discrete convolution involving 𝑟 -Whitney numbers of both kinds in terms of Bernoulli polynomials

∑ 𝑗𝑤𝑚,𝑟(𝑛 + 1, 𝑛 + 1 − 𝑗)𝑊𝑚.𝑟 𝑘

𝑗=1

(𝑛 + 𝑘

− 𝑗, 𝑛)

=

𝑚𝑘(𝐵𝑘+1(_{𝑚) − 𝐵}𝑟 𝑘+1(𝑛 + 1 +_𝑚))𝑟

𝑘 + 1

second kind with integral values of the parameters 𝑛 and 𝑚. In this paper an asymptotic formula for the 𝑟-Whitney numbers of the second kind with real parameters will be derived by using Hankel’s loop integral and the modified saddle-point method of Temme.

Theorem 1.1 For positive real numbers

𝑥 and 𝑦, the following asymptotic formula of the second kind 𝑟-Whitney numbers holds uniformly as

𝑦 → ∞ for 𝛿 < 𝑥 ≤ 𝑦:

𝑊𝛽,𝑟(𝑦, 𝑥)~𝛽𝑦−𝑥(𝑦𝑥)𝑒𝐴𝑓(𝑡0)(𝑣 +

𝑥)𝑦−𝑥_{, (1.1)}

where

𝑡0= 𝑦−𝑥

𝑣+𝑥, (1.2)

𝐴 = 𝜙(𝑢𝑜) − (𝑣 + 𝑥)𝑡0+ (𝑦 −

𝑥)𝑙𝑜𝑔𝑡0, (1.3)

𝑢0is the unique positive solution of the

equation

𝑣 + 𝑥𝑒𝑢

𝑒𝑢₋₁−

𝑦

𝑢= 0 (1.4)

𝜙(𝑢) = 𝑣𝑢 + 𝑥𝑙𝑜𝑔(𝑒𝑢_{− 1) −}

𝑦 log 𝑢, (1.5)

𝑓(𝑡0) = 1 𝑢0√

𝑡0(𝑣+𝑥)

𝜙′′_(𝑢 0)= √_{(𝑦−𝑣𝑢} 𝑥(𝑦−𝑥)

0)(𝑣+𝑥)(𝑢0−𝑡0)+𝑥𝑣𝑢0 (1.6)

where each logarithm is the principal logarithm.

The unique solution 𝑢0 maybe obtained using mathematica.

Deriving the Asymptotic Formula

Following Flajolet and Prodinger (1999), an integral

representation of 𝑟-Whitney numbers is given by

𝑊𝛽,𝑟(𝑦, 𝑥) = 𝑦! 2𝜋𝑖𝛽𝑥_𝑥!∫

𝑒𝑟𝑧(𝑒𝛽𝑧−1)𝑥 𝑧𝑦+1

𝐻 𝑑𝑧 (2.1)

where 𝑥 and 𝑦 are positive real numbers, 𝑦!

and 𝑥! are generalized factorials defined via the gamma function and H is a Hankel contour which starts at −∞, circles the origin and goes back to −∞.

With 𝑢 = 𝛽𝑧, 𝑣 =𝑟 𝛽, and

𝜙(𝑢) = 𝑣𝑢 + 𝑥𝑙𝑜𝑔(𝑒𝑢_{− 1) − 𝑦𝑙𝑜𝑔 𝑢, (2.2)}

(2.1) can be written as

𝑊𝛽,𝑟(𝑦, 𝑥) =

𝑦!

𝑥! 𝛽𝑥−𝑦_2𝜋𝑖∫ exp (𝜙(𝑢))

𝑑𝑢 𝑢

𝐻

.

(2.3)

The next lemma guarantees the existence of a unique real positive solution to the equation

𝜙′(𝑢) = 0.

Lemma 2.1. Suppose 𝑣, 𝑥 and 𝑦 are positive real numbers such that

0 < 𝑥 < 𝑦.

Then the equation

𝑣 + 𝑥𝑒𝑢

𝑒𝑢₋₁−

𝑦

𝑢= 0 (2.4)

ℎ𝑎𝑠 𝑎 𝑢𝑛𝑖𝑞𝑢𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛.

𝑃𝑟𝑜𝑜𝑓. Let

𝐺(𝑢) =𝑣𝑢 𝑥 +

𝑢𝑒𝑢

𝑒𝑢_{− 1}, 𝐻(𝑢) =

𝑦 𝑥.

Then equation (2.4) can be written 𝐺(𝑢) = 𝐻(𝑢). Note that 𝐺(𝑢) and

On the other hand, the function 𝐺(𝑢) is increasing in (0, +∞); 𝐺 → +∞ as (𝑢 → +∞); and 𝐺(𝑢) → 1 as (𝑢 → 0). Since

(𝑦 > 𝑥), the constant 𝑦/𝑥 is greater than 1. Hence, we can be certain that 𝐺(𝑢) and

𝐻(𝑢) will intersect at some point P whose

𝑥-coordinate is the desired solution to the equation.

We denote by 𝑢0 the unique solution to the equation 𝜙′_{(𝑢) = 0. Then}

𝑢0 satisfies (2.4). Observe also that

𝜙(𝑢)~(𝑥 − 𝑦)𝑙𝑜𝑔𝑢 𝑎𝑠 𝑢 → 0 𝑎𝑛𝑑

𝜙(𝑢)~(𝑣 + 𝑥)𝑢 𝑎𝑠 𝑢 → +∞.

These observations suggest the transformations 𝑡: 𝑢 → 𝑡(𝑢) defined by

𝜙(𝑢) = (𝑣 + 𝑥)𝑡 + (𝑥 − 𝑦)𝑙𝑜𝑔𝑡 + 𝐴, (2.5)

where A is a function independent of 𝑡

which is to be determined. The derivative of the right hand side of (2.5) is zero at 𝑡0= (𝑦 − 𝑥)/ (𝑣 + 𝑥), where 𝑣 + 𝑥 ≠

0.

Note that t0> 0 . We require for the mapping in (2.5) the corresponding points:

𝑢 = 0 ⇔ 𝑡 = 0, 𝑢 = 𝑢0 ⇔ 𝑡 = 𝑡0 and

𝑢 = +∞ ⇔ 𝑡 = +∞.

The substitutions 𝑢 = 𝑢0, 𝑡 = 𝑡0 in (2.5) give the value of the constant A to be

𝐴 = 𝜙(𝑢0) − (𝑣 + 𝑥)𝑡0+ (𝑦 − 𝑥)𝑙𝑜𝑔 𝑡0.

(2.6)

The transformation in (2.5) brings (2.3) in the form

𝑊𝛽,𝑟(𝑦, 𝑥)

= 𝑦! 𝑒

𝐴

𝑥! 𝛽𝑥−𝑦_2𝜋𝑖∫ 𝑒(𝑣+𝑥)𝑡 𝛾

𝑓(𝑡) 𝑑𝑡

𝑡𝑦−𝑥+1,

(2.7)

where

𝑓(𝑡) =

𝑢 𝑑𝑢

𝑑𝑡

and the contour 𝛾 is the image of 𝐻 under the transformation. The analyticity of the transformation defined in (2.5) is discussed in the next section. The contour 𝛾 which is still a Hankel contour, can be deformed so that it passes through the new saddle point

𝑡0. An approximation to 𝑊𝛽,𝑟(𝑦, 𝑥) is obtained by replacing 𝑓(𝑡) in equation (2.7) with the value of the function at 𝑡0, the point where we expect the major contribution to the integral occurs. Then we have

𝑊𝛽,𝑟(𝑦, 𝑥)~

𝑦!𝑒𝐴𝑓(𝑡0)

𝑥!𝛽𝑥−𝑦

1 2𝜋𝑖∫

𝑒(𝑣+𝑥)𝑡𝑑𝑡 𝑡𝑦−𝑥+1

𝛾 .

(2.9)

The integral in (2.9) will be evaluated using the following identity due to Hankel which is also known as Hankel’s loop integral (Olver, 1974),

1 2𝜋𝑖∫

𝑒𝑤

𝑤𝑥+1

𝐻 𝑑𝑤 =

1

ᴦ(𝑥+1), (2.10) where H is the same Hankel contour used in equation (2.1). Evaluation of the integral is as follows:

1 2𝜋𝑖∫

𝑒(𝑣+𝑥)𝑡 𝑡𝑦−𝑥+1 𝛾

𝑑𝑡 = (𝑣 + 𝑥)𝑦−𝑥 1

ᴦ(𝑦 − 𝑥 + 1)

=(𝑣 + 𝑥)

𝑦−𝑥

Hence, an approximation is given by

𝑊𝛽,𝑟(𝑦, 𝑥)~(𝑦_𝑥)𝛽𝑦−𝑥𝑒𝐴(𝑣 + 𝑥)𝑦−𝑥𝑓(𝑡0), (2.11)

as 𝑦 → ∞.

We now find an expression for

𝑓(𝑡0). Differentiating both sides of (2.5) bearing in mind that the transformation 𝑡 is implicitly defined, we have

𝑑𝑢

𝑑𝑡 = [(𝑣 + 𝑥) + 𝑥−𝑦

𝑡 ] . 1

𝜙′_(𝑢)

Substitution to (2.8) will give

𝑓(𝑡) =(𝑣 + 𝑥)(𝑡 − 𝑡0) 𝑢𝜙′_(𝑢) .

Note that at 𝑡 = 𝑡0, we have 𝑢 = 𝑢0 and

𝜙′_(𝑢

0) = 0. Hence,

𝑓(𝑡0) = 𝑙𝑖𝑚𝑡→𝑡₀

(𝑣 + 𝑥)(𝑡 − 𝑡₀) 𝑢𝜙′_(𝑢)

= _{𝑢𝜙′′(𝑢0)𝑑𝑢}(𝑣+𝑥) 𝑑𝑡

. (2.12))

From (2.8) an expression for 𝑑𝑢/𝑑𝑡

evaluated at 𝑢 = 𝑢0, and 𝑡 = 𝑡0 is obtained and is given by

𝑑𝑢

𝑑𝑡| 𝑢 = 𝑢0, 𝑡 = 𝑡0=

𝑢0𝑓(𝑡0)

𝑡0

.

Substitute to (2.12) to obtain

[𝑓(𝑡₀)]2 = (𝑣 + 𝑥)𝑡0 𝑢02𝜙′′(𝑢0)

,

which will give

𝑓(𝑡

) =

𝑢0

√

0)

Next we solve for

𝜙

(𝑢

).

From

(2.2),

𝜙′′_{(𝑢) =} −𝑥𝑒−𝑢 (1−𝑒−𝑢₎2+

𝑦

𝑢2. (2.14)

From (2.4),

𝑒

_{= 1 −}

.

Hence

𝜙′′_(𝑢

0) = −𝑥 (1 −

𝑥𝑢0

𝑦 − 𝑣𝑢0

) 1

[1 − (1 − 𝑥𝑢0

𝑦 − 𝑣𝑢0)] 2

+ 𝑦 𝑢02

= −𝑥 (𝑦 − 𝑣𝑢0− 𝑥𝑢0 𝑦 − 𝑣𝑢0

) 1

( 𝑥𝑢0

𝑦 − 𝑣𝑢0) 2+

𝑦 𝑢₀2

= −𝑥(𝑦 − 𝑣𝑢0− 𝑥𝑢0)

𝑦 − 𝑣𝑢0

𝑥𝑢02

+ 𝑦 𝑢02

= −𝑥(𝑦 − 𝑣𝑢0)

2

(𝑥𝑢0)2

+𝑥

2_𝑢

0(𝑦 − 𝑣𝑢0)

𝑥𝑢02

+ 𝑦 𝑢02

= −(𝑦 − 𝑣𝑢0)

2

𝑥𝑢₀2 +

𝑦 − 𝑣𝑢0

𝑢0

+ 𝑦 𝑢₀2

=𝑥𝑢0(𝑦−𝑣𝑢0)−(𝑦−𝑣𝑢0)2+ 𝑥𝑦

𝑥𝑢₀2 (2.15

)

Note that ϕ′′_(u

0) > 0. In fact, ϕ′′(u) > 0 on the interval (0, +∞). To see this, recall from (2.14),

𝜙′′_{(𝑢) =} −𝑥

𝑒𝑢_{(1 − 𝑒}−𝑢₎2+

𝑦

𝑢2

= −𝑥

𝑒𝑢_{− 2 + 𝑒}−𝑢

+ 𝑦

𝑢2. Since 𝑒𝑢_{+ 𝑒}−𝑢_{− 2 > 𝑢}2 _{for 𝑢 in (0, +∞),}

𝑥

𝑒𝑢_{− 2 + 𝑒}−𝑢<

𝑥 𝑢2<

𝑦

𝜙′′(𝑢) = 𝑦 𝑢2−

𝑥

𝑒𝑢_{− 2 + 𝑒}−𝑢> 0 and 𝜙′′(𝑢₀) > 0. Substituting (2.15) for

𝜙′′(𝑢₀) in (2.13) and using 𝑡0= (𝑦 −

𝑥)/(𝑣 + 𝑥), we have

𝑓(𝑡0) =_√

(𝑣 + 𝑥)𝑡0

𝑢02[

𝑥𝑢0(𝑦 − 𝑣𝑢0) − (𝑦 − 𝑣𝑢0)2+ 𝑥𝑦

𝑥𝑢₀2 ]

= √ 𝑥(𝑣 + 𝑥)𝑡0

(𝑦 − 𝑣𝑢0)[𝑥𝑢0− (𝑦 − 𝑣𝑢0)] + 𝑥𝑦

= √ 𝑥(𝑣 + 𝑥)𝑡0

(𝑦 − 𝑣𝑢0) {(𝑣 + 𝑥) [𝑢0− −_{𝑣 + 𝑥}𝑦 ]} + 𝑥𝑦

= √ 𝑥(𝑣 + 𝑥)𝑡0

(𝑦 − 𝑣𝑢0) {(𝑣 + 𝑥) [𝑢0− (𝑡0+_{𝑣 + 𝑥}𝑥 ]} + 𝑥𝑦

= √ 𝑥(𝑦 − 𝑥)

(𝑦 − 𝑣𝑢0){(𝑣 + 𝑥)(𝑢0− 𝑡0) − 𝑥} + 𝑥𝑦

𝑓(𝑡0) = √ 𝑥(𝑣 + 𝑥)𝑡0

(𝑦 − 𝑣𝑢0)(𝑣 + 𝑥)(𝑢0− 𝑡0) + 𝑥𝑣𝑢0

(2.16)

Exact and approximate values of 𝑊1,1=

(𝑦, 𝑥) for y=20 are given in Table 1. The exact values are computed using the triangular recurrence relation for 𝑊𝛽,𝑟=

The Analyticity of the Transformation The proof of the analyticity of 𝑓 entails showing that transformation t defined implicitly in (2.5) is analytic. In so doing the following implicit function theorem will be used. This theorem is quoted from [20] and the following discussion can be obtained from the corresponding discussion in [20].

Theorem 3.1 (Implicit Function Theorem).

Consider the equation

𝐹(𝑤, 𝑧) = 0

where 𝐹: ℂ × ℂ → ℂ is analytic in a neighborhood of (0,0) and 𝐹(0,0), 𝐷𝑤𝐹(0,0) ≠ 0. Then there exists 𝜀 > 0

such that for every 𝑧, |𝑧| < 𝜀 the equation

𝐹(𝑤, 𝑧) = 0 has a unique solution 𝑤(𝑧)

which is analytic in a neighborhood of zero.

We introduce a new function which we denote by 𝜏(𝑢) by writing

𝑡 = [𝑡0

𝑢0+ (𝑢 − 𝑢0)𝜏] 𝑢. (3.1)

This matches the points 𝑢 = 0, 𝑡 = 0 and

𝑢 = 𝑢0, 𝑡 = 𝑡0. We substitute (3.1) to (2.5) and obtain

𝜙(𝑢) = (𝑣 + 𝑥) [𝑡0 𝑢0

+ (𝑢 − 𝑢0)𝜏] 𝑢

+ (𝑥 − 𝑦)𝑙𝑜𝑔 [𝑡0 𝑢0

+ (𝑢 − 𝑢0)𝜏] 𝑢 + 𝐴

= (𝑣 + 𝑥)𝑢𝑡0 𝑢0

+ 𝑢(𝑣 + 𝑥)(𝑢 − 𝑢0)𝜏

+(𝑥 − 𝑦) log 𝑢 + (𝑥 − 𝑦)𝑙𝑜𝑔𝑡0 𝑢0

+ (𝑥 − 𝑦)𝑙𝑜𝑔 [1 +𝑢0𝜏(𝑢−𝑢0

𝑡0 ] + 𝐴. (3.2)

The last log term of the preceding equation can be expanded as

𝑙𝑜𝑔 [1 +𝑢0𝜏(𝑢 − 𝑢0 𝑡0

]

= ∑(−1)

𝑠+1

𝑠

∞

𝑠=1

(𝑢0𝜏(𝑢 − 𝑢0) 𝑡0

)

𝑠

=𝑢0𝜏(𝑢 − 𝑢0 𝑡0

− ∑(−1)

𝑠

𝑠

∞

𝑠=2

(𝑢0𝜏(𝑢 − 𝑢0) 𝑡0

)

𝑠

and so

(𝑥 − 𝑦)𝑙𝑜𝑔 [1 +𝑢0𝜏(𝑢 − 𝑢0) 𝑡0

]

= (𝑥 − 𝑦) [𝑢0𝜏(𝑢

− 𝑢0)

𝑣 + 𝑥 𝑦 − 𝑥−

∑(−1)

𝑠

𝑠

∞

𝑠 =2

(𝑢0𝜏(𝑢 − 𝑢0) 𝑡0

)

𝑠

]

= −𝑢0(𝑣 + 𝑥)(𝑢 − 𝑢0)𝜏 +

∑(−1)

𝑠

𝑠

∞

𝑠=2

(𝑢0(𝑣 + 𝑥)(𝑢 − 𝑢0)𝜏)𝑠

(𝑦 − 𝑥)𝑠−1

Thus,

𝜙(𝑢) = (𝑣 + 𝑥)𝑢𝑡0 𝑢0

+ (𝑣 + 𝑥)(𝑢 − 𝑢0)2𝜏

+ (𝑥 − 𝑦)𝑙𝑜𝑔𝑢 + (𝑥 − 𝑦)𝑙𝑜𝑔𝑡0

𝑢0

+

∑(−1)

𝑠

𝑠

∞

𝑠=2

[𝑢0(𝑣 + 𝑥)(𝑢 − 𝑢0)𝜏]𝑠

(𝑦 − 𝑥)𝑠−1 + 𝐴.

𝑄(𝑢) = 𝜙(𝑢) − (𝑣 + 𝑥)𝑢𝑡0

𝑢₀− (𝑥 −

𝑦) log 𝑢 − (𝑥 − 𝑦) log𝑡0

𝑢₀− 𝐴. (3.3) Then

𝑄(𝑢) − (𝑣 + 𝑥)(𝑢 − 𝑢0)2𝜏

− ∑(−1)

𝑠

𝑠

∞

𝑠=2

[𝑢0(𝑣 + 𝑥)(𝑢 − 𝑢0)𝜏]𝑠

(𝑦 − 𝑥)𝑠−1 = 0

𝑄(𝑢)

(𝑣 + 𝑥)(𝑢 − 𝑢0)2

− 𝜏

− ∑(−1)

𝑠

𝑠

∞

𝑠=2

𝑢₀𝑠(𝑣 + 𝑥)𝑠−1_{(𝑢 − 𝑢}

0)𝑠−2𝜏𝑠

(𝑦 − 𝑥)𝑠−1

= 0

𝑄(𝑢)

(𝑣 + 𝑥)(𝑢 − 𝑢₀)2− 𝜏

−𝑢0

2_{(𝑣 + 𝑥)𝜏}2

𝑦 − 𝑥 ∑

(−1)𝑠

𝑠

∞

𝑠=2

(𝑢0(𝑣 + 𝑥)(𝑢 − 𝑢0)𝜏

𝑦 − 𝑥 )

𝑠−2

= 0

𝑄(𝑢) (𝑣 + 𝑥)(𝑢 − 𝑢0)2

− 𝜏 −

𝑢02𝜏2

𝑡0

∑(−1)

𝑠

𝑠

∞

𝑠=2

(𝑢0(𝑢 − 𝑢0)𝜏 𝑡0

)

𝑠−2

= 0

(3.4)

From (2.6) it follows that

𝑄(𝑢0) = 𝜙(𝑢0) − (𝑣 + 𝑥)𝑢0

𝑡0

𝑢0

−(𝑥 − 𝑦) log 𝑢0− (𝑥 − 𝑦) log 𝑡0+

(𝑥 − 𝑦) log 𝑢0− (𝜙(𝑢0) − (𝑣 − 𝑥)𝑡0)

+(𝑦 − 𝑥) log 𝑡0) = 0

(3.5)

and

𝑄′(𝑢₀) = 𝜙′_(𝑢

0) − (𝑣 − 𝑥) 1 𝑢0

𝑦−𝑥 𝑣+𝑥+

𝑦−𝑥

𝑢0 = 0 (3.6)

since 𝜙′(𝑢₀) = 0.

From equations (3.5) and (3.6) it follows that the function 𝑄(𝑢)

(𝑣+𝑥)(𝑢−𝑢0)2 has a

removable singularity at 𝑢 = 𝑢0, thus 𝑄(𝑢)

(𝑣+𝑥)(𝑢−𝑢0)2 is analytic at 𝑢 = 𝑢0. Also,

the series on the left hand side of (3.4) represents an analytic function for values of 𝑢, 𝜏 and 𝑡0 close to zero.

Moreover, from (3.3)

𝑄(𝑢) = 𝑢(𝜈 + 𝑥)(𝑢 − 𝑢0)𝜏 +

(𝑥 − 𝑦)𝑙𝑜𝑔 [1 +𝑢0𝜏(𝑢 − 𝑢0) 𝑡0

], (3.7)

from which 𝑄(𝜏 = 0) = 0.

Let 𝑤 = 𝜏, 𝑧 = (𝑢, 𝑡) and 𝐹(𝑤, 𝑧)

denote the left hand side of (3.4). Then

𝐹(𝑤, 𝑧) is analytic for values of 𝑤 and 𝑧

close to zero. Also,

𝐹(0, 𝑧) = 0, ∀𝑧.

Taking the derivative with respect to 𝑤,

𝐷𝑤𝐹(𝑤, 𝑧) = 𝐷𝑤𝑄(𝑢) − 1 −

∑(−1)𝑠 ∞

𝑠=2

𝑢₀𝑠_{(𝑢 − 𝑢} 0)𝑠−2

𝜏𝑠−1 𝑡₀𝑠−1.

𝐷𝑤𝑄(𝑢) = 𝑢(𝜈 + 𝑥)(𝑢 − 𝑢0)

+(𝑥 − 𝑦) [ 𝑢0

𝑡0(𝑢 − 𝑢0)

(1 +𝑢0𝜏(𝑢 − 𝑢_𝑡 0)

0 )

]

Evaluating at 𝑤 = 0,

𝐷

𝑄(𝑢)|

= 𝑢(𝜈 + 𝑥)(𝑢 − 𝑢

)

+ (𝑥 − 𝑦)

𝑢

𝑡

(𝑢 − 𝑢

).

Thus,

𝐷

𝐹(𝑤, 𝑧)|

= 𝐷

𝑄(𝑢)|

− 1.

Notice that evaluating

𝐷

𝑄(𝑢)|

at

𝑢 = 0

and

𝑡

= 0

, will yield

0 −

(𝑦−𝑥)

𝑢

(𝑢 − 𝑢

)(𝜈 + 𝑥) = 0

since

𝑢

= 0

when

𝑡

= 0

. Thus,

𝐷

𝐹(0,0) ≠ 0.

It follows from the Implicit Function Theorem that 𝜏, and hence, 𝑡, is analytic in

𝑢 and 𝑡0 for values of these variables close to zero. Consequently, the function 𝑓 is analytic at 𝑡 = 0, and 𝑡 = 𝑡0, also when 𝑡0 (that is, 𝑢0) tends to zero.

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