Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine

Wei-Juan Pei; Bai-Ni Guo

doi:10.1515/math-2024-0095

Article Open Access

Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine

Wei-Juan Pei and Bai-Ni Guo

Published/Copyright: December 2, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

Keywords: Maclaurin series expansion; Qi’s normalized remainder; Bernoulli number; monotonicity; logarithmic convexity; monotonicity rule; derivative formula; cosine; ratio

MSC 2010: 41A58; 11B68; 15A15; 26A06; 26A09; 26A51; 33B10

This article is dedicated to Professor Dr. Feng Qi for his retirement in 2025.

1 Motivations

Let

(1) F ( t ) = ln 2 ( 1 − cos t ) t 2 , 0 < ∣ t ∣ < 2 π , 0 , t = 0

for t ∈ ( − 2 π , 2 π ) and

R ( t ) = F ( t ) ln cos t , 0 < ∣ t ∣ < π 2 , 1 6 , t = 0 , 0 , t = ± π 2

for t ∈ − π 2 , π 2 . In [1], the even function F ( t ) was expanded into a Maclaurin series about t = 0 and the even function R ( t ) was proved to be decreasing from 0 , π 2 onto 0 , 1 6 .

In [2–4], Qi and his coauthors introduced the normalized remainder

CosR n ( t ) = ( − 1 ) n ( 2 n ) ! t 2 n cos t − ∑ k = 0 n − 1 ( − 1 ) k t 2 k ( 2 k ) ! , 0 < ∣ t ∣ < 2 π , 1 , t = 0

and its logarithm F n ( t ) = ln CosR n ( t ) for n ≥ 0 , as well as the authors investigated these two even functions.

It is easy to see that F 0 ( t ) = ln cos t and F 1 ( t ) = F ( t ) , which are the logarithms of the first two normalized remainders CosR 0 ( t ) and CosR 1 ( t ) .

In [1, Remark 2], Milovanović pointed out and the authors verified that

(2) F ( t ) = ln CosR 1 ( t ) = − ∑ n = 1 ∞ ∣ B 2 n ∣ n t 2 n ( 2 n ) ! , ∣ t ∣ < 2 π ,

where the Bernoulli numbers B k are generated [5, Chapter 1] by

t e t − 1 = ∑ k = 0 ∞ B k t k k ! = 1 − t 2 + ∑ k = 1 ∞ B 2 k t 2 k ( 2 k ) ! = 1 − t 2 + t 2 12 − t 4 720 + t 6 30240 − t 8 1209600 + … , ∣ t ∣ < 2 π .

In [1, Remark 3], based on the Maclaurin series expansion (2), Qi and his coauthor constructed a positive and even function

(3) H m ( t ) = − m + 1 ∣ B 2 m + 2 ∣ ( 2 m + 2 ) ! t 2 m + 2 F ( t ) + ∑ n = 1 m ∣ B 2 n ∣ n t 2 n ( 2 n ) ! , 0 < ∣ t ∣ < 2 π , 1 , t = 0

for m ∈ N . As done in [2–4, 6–8] and [9, Section 7], we call the quantity H m ( t ) for m ∈ N Qi’s normalized remainder of the Maclaurin series expansion (2) of the function F ( t ) defined by (1).

In [1, Remark 3], Qi and his coauthor proposed the following two problems:

discuss the logarithmic convexity or logarithmic concavity of Qi’s normalized remainder H m ( t ) on ( 0 , 2 π ) ;
expand the logarithm ln H m ( t ) into a Maclaurin series about t = 0 .

Now, we pose two more problems: For given m ∈ N ,

discuss the monotonicity of the ratio H m + 1 ( t ) H m ( t ) in t ∈ ( − 2 π , 2 π ) ,
discuss the monotonicity of the ratio ln H m + 1 ( t ) ln H m ( t ) in t ∈ ( − 2 π , 2 π ) .

In this article, we will give confirmative solutions to the first three problems.

In Section 2, we prepare three lemmas. In Section 3, we state and prove the logarithmic convexity of H m ( t ) in t ∈ ( − 2 π , 2 π ) . In Section 4, we expand the logarithm ln H m ( t ) into a Maclaurin series about t = 0 . In Section 5, we discuss the monotonicity of the ratio H m + 1 ( t ) H m ( t ) in t ∈ ( − 2 π , 2 π ) . In Section 6, we connect our main results with the generalized hypergeometric function F 2 1 .

2 Lemmas

For smoothly proceeding, we recall the following three lemmas which are effective and applicable extensively in mathematical sciences.

Lemma 1

(Monotonicity rule for the quotient of two power series [10–12]) Let α ℓ and β ℓ for ℓ ∈ { 0 } ∪ N be real sequences and the Maclaurin series

P ( t ) = ∑ ℓ = 0 ∞ α ℓ t ℓ and Q ( t ) = ∑ ℓ = 0 ∞ β ℓ t ℓ

converge on ( − ρ , ρ ) for some scalar ρ > 0 . If β ℓ > 0 and the sequence α ℓ β ℓ increases in ℓ ≥ 0 , then the function t ↦ P ( t ) Q ( t ) increases on ( 0 , ρ ) .

Lemma 2

The sequence

ℓ ℓ + 1 ( 2 ℓ ) ! ( 2 ℓ + 2 ) ! B 2 ℓ + 2 B 2 ℓ

is increasing in ℓ ≥ 0 .

Proof

In the proof of [13, Theorem 1.1], the identity

∣ B 2 ( ℓ + 1 ) ∣ ∣ B 2 ℓ ∣ = 1 2 π 2 ( 2 ℓ + 1 ) ( ℓ + 1 ) ζ ( 2 ℓ + 2 ) ζ ( 2 ℓ )

for ℓ ∈ N was derived, where the Riemann zeta function ζ ( z ) can be defined by the series ζ ( z ) = ∑ n = 1 ∞ 1 n z under the condition ℜ ( z ) > 1 and by analytic continuation elsewhere. Hence, we have

ℓ ℓ + 1 ( 2 ℓ ) ! ( 2 ℓ + 2 ) ! ∣ B 2 ℓ + 2 ∣ ∣ B 2 ℓ ∣ = 1 4 π 2 ℓ ℓ + 1 ζ ( 2 ℓ + 2 ) ζ ( 2 ℓ ) , ℓ ≥ 1 .

The sequence ℓ ℓ + 1 is increasing in ℓ ≥ 1 . Therefore, it suffices to show that the function ζ ( ℓ + 2 ) ζ ( ℓ ) = ζ ( ℓ + 1 ) ζ ( ℓ ) ζ ( ℓ + 2 ) ζ ( ℓ + 1 ) is increasing in ℓ ≥ 2 . So, it is enough to verify that ζ ( ℓ + 1 ) ζ ( ℓ ) is increasing in ℓ > 1 . This is equivalent to ζ ( ℓ + 1 ) ζ ( ℓ ) ≤ ζ ( ℓ + 2 ) ζ ( ℓ + 1 ) , that is, ζ ( ℓ ) ζ ( ℓ + 2 ) ≥ [ ζ ( ℓ + 1 ) ] 2 for ℓ > 1 . This inequality was established in [14, Section 3] or it can be derived from the conclusion in [14, Section 3] that the reciprocal 1 ζ ( t ) is concave on ( 1 , ∞ ) . The proof of Lemma 2 is complete.□

Remark 1

We guess that the function ζ ( t ) t is decreasing and logarithmically convex in t > 1 .

Lemma 3

[15, p. 40, Exercise 5] For the integer n ≥ 0 and for two nth differentiable functions p ( t ) and q ( t ) ≠ 0 , let

W ( n + 1 ) × ( n + 1 ) ( t ) = P ( n + 1 ) × 1 ( t ) Q ( n + 1 ) × n ( t ) ( n + 1 ) × ( n + 1 )

and let ∣ W ( n + 1 ) × ( n + 1 ) ( t ) ∣ denote the determinant of the ( n + 1 ) × ( n + 1 ) matrix, where the ( n + 1 ) × 1 matrix P ( n + 1 ) × 1 ( t ) is of the elements p ℓ , 1 ( t ) = p ( ℓ − 1 ) ( t ) for 1 ≤ ℓ ≤ n + 1 , and the ( n + 1 ) × n matrix Q ( n + 1 ) × n ( t ) is of the elements q ℓ , j ( t ) = ℓ − 1 j − 1 q ( ℓ − j ) ( t ) for 1 ≤ ℓ ≤ n + 1 and 1 ≤ j ≤ n . Then, the nth derivative of the quotient p ( t ) q ( t ) can be computed by the determinantal formula

(4) d n d t n p ( t ) q ( t ) = ( − 1 ) n ∣ W ( n + 1 ) × ( n + 1 ) ( t ) ∣ q n + 1 ( t ) , n ≥ 0 .

3 Logarithmic convexity

In this section, with the aid of Lemmas 1 and 2, we prove that Qi’s normalized remainder H m ( t ) for m ∈ N is logarithmically convex in t ∈ ( − 2 π , 2 π ) .

Theorem 1

For m ∈ N , Qi’s normalized remainder H m ( t ) is

even in t ∈ ( − 2 π , 2 π ) ,
increasing in t ∈ ( 0 , 2 π ) and decreasing in t ∈ ( − 2 π , 0 ) ,
logarithmically convex in t ∈ ( − 2 π , 2 π ) .

Proof

We write

(5) H m ( t ) = ∑ n = 0 ∞ m + 1 m + n + 1 ( 2 m + 2 ) ! ( 2 m + 2 n + 2 ) ! ∣ B 2 m + 2 n + 2 ∣ ∣ B 2 m + 2 ∣ t 2 n , ∣ t ∣ < 2 π .

This means that Qi’s normalized remainder H m ( t ) is even in t ∈ ( − 2 π , 2 π ) , increasing in t ∈ ( 0 , 2 π ) , and decreasing in t ∈ ( − 2 π , 0 ) .

Taking the logarithm and differentiating on both sides of (5) yield

[ ln H m ( t ) ] ′ x = ∑ n = 0 ∞ m + 1 m + n + 2 ( 2 m + 2 ) ! ( 2 m + 2 n + 4 ) ! ∣ B 2 m + 2 n + 4 ∣ ∣ B 2 m + 2 ∣ ( 2 n + 2 ) t 2 n ∑ n = 0 ∞ m + 1 m + n + 1 ( 2 m + 2 ) ! ( 2 m + 2 n + 2 ) ! ∣ B 2 m + 2 n + 2 ∣ ∣ B 2 m + 2 ∣ t 2 n .

In order to use Lemma 1, we consider the increasing property of the quotient

m + 1 m + n + 2 ( 2 m + 2 ) ! ( 2 m + 2 n + 4 ) ! ∣ B 2 m + 2 n + 4 ∣ ∣ B 2 m + 2 ∣ ( 2 n + 2 ) m + 1 m + n + 1 ( 2 m + 2 ) ! ( 2 m + 2 n + 2 ) ! ∣ B 2 m + 2 n + 2 ∣ ∣ B 2 m + 2 ∣ = m + n + 1 m + n + 2 ( 2 m + 2 n + 2 ) ! ( 2 m + 2 n + 4 ) ! ∣ B 2 m + 2 n + 4 ∣ ∣ B 2 m + 2 n + 2 ∣ ( 2 n + 2 )

in n ≥ 0 . Equivalently, we consider the increasing property of the sequence

ℓ + 1 ℓ + 2 ( 2 ℓ + 2 ) ! ( 2 ℓ + 4 ) ! ∣ B 2 ℓ + 4 ∣ ∣ B 2 ℓ + 2 ∣ ( ℓ − m + 1 )

in ℓ ≥ m ∈ N . It is sufficient to show the increasing property of the sequence

ℓ + 1 ℓ + 2 ( 2 ℓ + 2 ) ! ( 2 ℓ + 4 ) ! ∣ B 2 ℓ + 4 ∣ ∣ B 2 ℓ + 2 ∣

in ℓ ≥ 1 . This has been proved in Lemma 2. Accordingly, by virtue of Lemma 1, we see that the function [ ln H m ( t ) ] ′ x , and then [ ln H m ( t ) ] ′ is increasing in t ∈ ( 0 , 2 π ) . Consequently, Qi’s normalized remainder H m ( t ) is logarithmically convex on ( 0 , 2 π ) . The proof of Theorem 1 is complete.□

4 Maclaurin series expansion

In this section, based on the proof of Theorem 1, by virtue of the higher order derivative formula (4), we expand the logarithm ln H m ( t ) into a Maclaurin series about t = 0 .

Theorem 2

Let

ω ℓ = ∣ B ℓ ∣ ℓ × ℓ ! , ℓ ≥ 1

and M m , 2 ℓ = θ j , k ( m ) ( 2 ℓ ) × ( 2 ℓ ) , where

θ j , 1 = j ! ω j + 2 m + 2 , 1 ≤ j ≤ 2 ℓ ; θ j , k = 0 , 2 ≤ k − j ≤ 2 ℓ − 1 ; θ j , k = j − 1 k − 2 ( j − k + 1 ) ! ω j − k + 2 m + 3 , − 1 ≤ j − k .

Then, the logarithm of Qi’s normalized remainder H m ( t ) for m ∈ N can be expanded into

ln H m ( t ) = − ∑ ℓ = 1 ∞ M m , 2 ℓ ω 2 ( m + 1 ) 2 ℓ t 2 ℓ ( 2 ℓ ) ! , ∣ t ∣ < 2 π .

Proof

For fixed m ∈ N , let

q ( t ) = ∑ n = 0 ∞ m + 1 m + n + 1 ( 2 m + 2 ) ! ( 2 m + 2 n + 2 ) ! ∣ B 2 m + 2 n + 2 ∣ ∣ B 2 m + 2 ∣ t 2 n = ∑ n = 0 ∞ ω 2 ( m + n + 1 ) ω 2 ( m + 1 ) t 2 n .

Then, for ℓ ≥ 0 , we have

[ ln H m ( t ) ] ( ℓ + 1 ) = q ′ ( t ) q ( t ) ( ℓ ) , q ( 2 ℓ + 1 ) ( 0 ) = 0 ,

and

q ( 2 ℓ ) ( 0 ) = ( 2 ℓ ) ! ω 2 ( ℓ + m + 1 ) ω 2 ( m + 1 ) .

Utilizing formula (4), we acquire

lim t → 0 [ ln H m ( t ) ] ( 2 ℓ + 1 ) = lim t → 0 q ′ ( t ) q ( t ) ( 2 ℓ ) = ( − 1 ) 2 ℓ q 2 ℓ + 1 ( 0 ) q ′ ( 0 ) 0 0 q ( 0 ) 0 0 ⋯ 0 q ″ ( 0 ) 1 0 q ′ ( 0 ) 1 1 q ( 0 ) 0 ⋯ 0 q ( 3 ) ( 0 ) 2 0 q ″ ( 0 ) 2 1 q ′ ( 0 ) 2 2 q ( 0 ) ⋯ 0 q ( 4 ) ( 0 ) 3 0 q ( 3 ) ( 0 ) 3 1 q ″ ( 0 ) 3 2 q ′ ( 0 ) ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ q ( 2 ℓ − 2 ) ( 0 ) 2 ℓ − 3 0 q ( 2 ℓ − 3 ) ( 0 ) 2 ℓ − 3 1 q ( 2 ℓ − 4 ) ( 0 ) 2 ℓ − 3 2 q ( 2 ℓ − 5 ) ( 0 ) ⋯ 0 q ( 2 ℓ − 1 ) ( 0 ) 2 ℓ − 2 0 q ( 2 ℓ − 2 ) ( 0 ) 2 ℓ − 2 1 q ( 2 ℓ − 3 ) ( 0 ) 2 ℓ − 2 2 q ( 2 ℓ − 4 ) ( 0 ) ⋯ 0 q ( 2 ℓ ) ( 0 ) 2 ℓ − 1 0 q ( 2 ℓ − 1 ) ( 0 ) 2 ℓ − 1 1 q ( 2 ℓ − 2 ) ( 0 ) 2 ℓ − 1 2 q ( 2 ℓ − 3 ) ( 0 ) ⋯ 2 ℓ − 1 2 ℓ − 1 q ( 0 ) q ( 2 ℓ + 1 ) ( 0 ) 2 ℓ 0 q ( 2 ℓ ) ( 0 ) 2 ℓ 1 q ( 2 ℓ − 1 ) ( 0 ) 2 ℓ 2 q ( 2 ℓ − 2 ) ( 0 ) ⋯ 2 ℓ 2 ℓ − 1 q ′ ( 0 ) = 0 0 0 q ( 0 ) 0 0 ⋯ 0 q ″ ( 0 ) 0 1 1 q ( 0 ) 0 ⋯ 0 0 2 0 q ″ ( 0 ) 0 2 2 q ( 0 ) ⋯ 0 q ( 4 ) ( 0 ) 0 3 1 q ″ ( 0 ) 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ q ( 2 ℓ − 2 ) ( 0 ) 0 2 ℓ − 3 1 q ( 2 ℓ − 4 ) ( 0 ) 0 ⋯ 0 0 2 ℓ − 2 0 q ( 2 ℓ − 2 ) ( 0 ) 0 2 ℓ − 2 2 q ( 2 ℓ − 4 ) ( 0 ) ⋯ 0 q ( 2 ℓ ) ( 0 ) 0 2 ℓ − 1 1 q ( 2 ℓ − 2 ) ( 0 ) 0 ⋯ 2 ℓ − 1 2 ℓ − 1 q ( 0 ) 0 2 ℓ 0 q ( 2 ℓ ) ( 0 ) 0 2 ℓ 2 q ( 2 ℓ − 2 ) ( 0 ) ⋯ 0 .

Since H m ( t ) is even, we arrive at

(6) 0 0 0 q ( 0 ) 0 0 ⋯ 0 q ″ ( 0 ) 0 1 1 q ( 0 ) 0 ⋯ 0 0 2 0 q ″ ( 0 ) 0 2 2 q ( 0 ) ⋯ 0 q ( 4 ) ( 0 ) 0 3 1 q ″ ( 0 ) 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ q ( 2 ℓ − 2 ) ( 0 ) 0 2 ℓ − 3 1 q ( 2 ℓ − 4 ) ( 0 ) 0 ⋯ 0 0 2 ℓ − 2 0 q ( 2 ℓ − 2 ) ( 0 ) 0 2 ℓ − 2 2 q ( 2 ℓ − 4 ) ( 0 ) ⋯ 0 q ( 2 ℓ ) ( 0 ) 0 2 ℓ − 1 1 q ( 2 ℓ − 2 ) ( 0 ) 0 ⋯ 2 ℓ − 1 2 ℓ − 1 q ( 0 ) 0 2 ℓ 0 q ( 2 ℓ ) ( 0 ) 0 2 ℓ 2 q ( 2 ℓ − 2 ) ( 0 ) ⋯ 0 = 0

for ℓ ≥ 0 and m ∈ N , where we used the fact B 2 ℓ + 1 = 0 for ℓ ≥ 1 .

Similarly, employing formula (4), we arrive at

lim t → 0 [ ln H m ( t ) ] ( 2 ℓ ) = lim t → 0 q ′ ( t ) q ( t ) ( 2 ℓ − 1 ) = ( − 1 ) 2 ℓ − 1 q 2 ℓ ( 0 ) × q ′ ( 0 ) 0 0 q ( 0 ) 0 0 ⋯ 0 q ″ ( 0 ) 1 0 q ′ ( 0 ) 1 1 q ( 0 ) 0 ⋯ 0 q ( 3 ) ( 0 ) 2 0 q ″ ( 0 ) 2 1 q ′ ( 0 ) 2 2 q ( 0 ) ⋯ 0 q ( 4 ) ( 0 ) 3 0 q ( 3 ) ( 0 ) 3 1 q ″ ( 0 ) 3 2 q ′ ( 0 ) ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ q ( 2 ℓ − 3 ) ( 0 ) 2 ℓ − 4 0 q ( 2 ℓ − 4 ) ( 0 ) 2 ℓ − 4 1 q ( 2 ℓ − 5 ) ( 0 ) 2 ℓ − 4 2 q ( 2 ℓ − 6 ) ( 0 ) ⋯ 0 q ( 2 ℓ − 2 ) ( 0 ) 2 ℓ − 3 0 q ( 2 ℓ − 3 ) ( 0 ) 2 ℓ − 3 1 q ( 2 ℓ − 4 ) ( 0 ) 2 ℓ − 3 2 q ( 2 ℓ − 5 ) ( 0 ) ⋯ 0 q ( 2 ℓ − 1 ) ( 0 ) 2 ℓ − 2 0 q ( 2 ℓ − 2 ) ( 0 ) 2 ℓ − 2 1 q ( 2 ℓ − 3 ) ( 0 ) 2 ℓ − 2 2 q ( 2 ℓ − 4 ) ( 0 ) ⋯ 2 ℓ − 2 2 ℓ − 2 q ( 0 ) q ( 2 ℓ ) ( 0 ) 2 ℓ − 1 0 q ( 2 ℓ − 1 ) ( 0 ) 2 ℓ − 1 1 q ( 2 ℓ − 2 ) ( 0 ) 2 ℓ − 1 2 q ( 2 ℓ − 3 ) ( 0 ) ⋯ 2 ℓ − 1 2 ℓ − 2 q ′ ( 0 ) = − 1 ω 2 ( m + 1 ) 2 ℓ 0 ω 2 ( m + 1 ) 0 2 ! ω 2 ( m + 2 ) 0 ω 2 ( m + 1 ) 0 2 ! ω 2 ( m + 2 ) 0 4 ! ω 2 ( m + 3 ) 0 3 1 2 ! ω 2 ( m + 2 ) ⋮ ⋮ ⋮ 0 ( 2 ℓ − 4 ) ! ω 2 ( ℓ + m − 1 ) 0 ( 2 ℓ − 2 ) ! ω 2 ( ℓ + m ) 0 2 ℓ − 3 1 ( 2 ℓ − 4 ) ! ω 2 ( ℓ + m − 1 ) 0 ( 2 ℓ − 2 ) ! ω 2 ( ℓ + m ) 0 ( 2 ℓ ) ! ω 2 ( ℓ + m + 1 ) 0 2 ℓ − 1 1 ( 2 ℓ − 2 ) ! ω 2 ( ℓ + m ) 0 ⋯ 0 0 0 0 ⋯ 0 0 0 ω 2 ( m + 1 ) ⋯ 0 0 0 0 ⋯ 0 0 0 ⋮ ⋱ ⋮ ⋮ ⋮ 2 ℓ − 4 2 ( 2 ℓ − 6 ) ! ω 2 ( ℓ + m − 2 ) ⋯ ω 2 ( m + 1 ) 0 0 0 ⋯ 0 ω 2 ( m + 1 ) 0 2 ℓ − 2 2 ( 2 ℓ − 4 ) ! ω 2 ( ℓ + m − 1 ) ⋯ 2 ℓ − 2 2 ℓ − 4 2 ! ω 2 ( m + 2 ) 0 ω 2 ( m + 1 ) 0 ⋯ 0 2 ℓ − 1 2 ℓ − 3 2 ! ω 2 ( m + 2 ) 0 .

Consequently, the series expansion of ln H m ( t ) about t = 0 is

ln H m ( t ) = ∑ ℓ = 0 ∞ lim t → 0 [ ln H m ( t ) ] ( ℓ ) t ℓ ℓ ! = ∑ ℓ = 0 ∞ lim t → 0 [ ln H m ( t ) ] ( 2 ℓ ) t 2 ℓ ( 2 ℓ ) ! = − ∑ ℓ = 1 ∞ M m , 2 ℓ ω 2 ( m + 1 ) 2 ℓ t 2 ℓ ( 2 ℓ ) ! .

The proof of Theorem 2 is complete.□

Remark 2

Can one verify equality (6) only by operations of linear algebra?

Remark 3

Taking ℓ = 1 , 2 and m = 2 leads to

M 2 , 2 = 0 ω 6 2 ! ω 8 0 = − 2 ∣ B 6 ∣ 6 × 6 ! ∣ B 8 ∣ 8 × 8 ! = − 1 877879296000

and

M 2 , 4 = 0 ω 6 0 0 2 ! ω 8 0 ω 6 0 0 2 ! ω 8 0 ω 6 4 ! ω 10 0 3 1 2 ! ω 8 0 = 12 ω 6 2 ( ω 8 2 − 2 ω 6 ω 10 ) = − 1 222438873983090688000000 .

Then, we arrive at

(7) ln H 2 ( t ) = − M 2 , 2 ω 6 2 t 2 2 ! − M 2 , 4 ω 6 4 t 4 4 ! − … = 3 160 t 2 + 343 1689600 t 4 + … , ∣ t ∣ < 2 π .

On the other hand, from the definitions in (1) and (3) and by the Maclaurin series expression (5), we have

H 2 ( t ) = − 90720 t 6 t 2 12 + t 4 1440 + ln 2 ( 1 − cos t ) t 2 = 90720 ∑ n = 0 ∞ ( 2 n ) ! n + 3 ∣ B 2 n + 6 ∣ ( 2 n + 6 ) ! t 2 n ( 2 n ) ! , ∣ t ∣ < 2 π

and the derivatives

H 2 ( 2 n + 1 ) ( 0 ) = 0 , H 2 ( 2 n ) ( 0 ) = 90720 ( 2 n ) ! n + 3 ∣ B 2 n + 6 ∣ ( 2 n + 6 ) ! , n ≥ 0 .

In particular,

H 2 ( 0 ) = 1 , H 2 ″ ( 0 ) = 3 80 , H 2 ( 4 ) = 1 110 .

Accordingly, we obtain

lim t → 0 [ ln H 2 ( t ) ] ″ = lim t → 0 H 2 ( t ) H 2 ″ ( t ) − [ H ′ ( t ) ] 2 H 2 2 ( t ) = 3 80

and

lim t → 0 [ ln H 2 ( t ) ] ( 4 ) = lim t → 0 H 2 ( t ) [ H ( t ) H ( 4 ) ( t ) − 3 [ H ″ ( t ) ] 2 ] − 6 [ H ′ ( t ) ] 4 − 4 H 2 ( t ) H ( 3 ) ( t ) H ′ ( t ) + 12 H ( t ) [ H ′ ( t ) ] 2 H ″ ( t ) H 4 ( t ) = 343 70400 .

As a result, we acquire

ln H 2 ( t ) = ∑ ℓ = 0 ∞ lim t → 0 [ ln H 2 ( t ) ] ( ℓ ) t ℓ ℓ ! = ∑ ℓ = 1 ∞ lim t → 0 [ ln H 2 ( t ) ] ( 2 ℓ ) t 2 ℓ ( 2 ℓ ) ! = 3 80 t 2 2 ! + 343 70400 t 4 4 ! + … = 3 160 t 2 + 343 1689600 t 4 + … , ∣ t ∣ < 2 π .

This series expansion coincides with the one in (7). This implies that Theorem 2 is true.

5 Monotonicity of ratio between normalized remainders

In this section, we discuss the monotonicity of the ratio H m + 1 ( t ) H m ( t ) in t ∈ ( − 2 π , 2 π ) for given m ∈ N .

Theorem 3

For given m ∈ N , the ratio H m + 1 ( t ) H m ( t ) is increasing in t ∈ ( 0 , 2 π ) and is decreasing in t ∈ ( − 2 π , 0 ) . Consequently, the inequality

H m + 1 ( t ) ≥ H m ( t )

is sound for m ∈ N and t ∈ ( − 2 π , 2 π ) .

Proof

From the series representation (5), we deduce

H m + 1 ( t ) H m ( t ) = ∑ n = 0 ∞ m + 2 n + m + 2 ∣ B 2 n + 2 m + 4 ∣ ∣ B 2 m + 4 ∣ 1 2 n + 2 m + 4 2 m + 4 t 2 n ( 2 n ) ! ∑ n = 0 ∞ m + 1 n + m + 1 ∣ B 2 n + 2 m + 2 ∣ ∣ B 2 m + 2 ∣ 1 2 n + 2 m + 2 2 m + 2 t 2 n ( 2 n ) !

for ∣ t ∣ < 2 π and m ∈ N . The ratio between coefficients of t 2 n in the series in the numerator and denominator is

(8) m + 2 n + m + 2 ∣ B 2 n + 2 m + 4 ∣ ∣ B 2 m + 4 ∣ 1 2 n + 2 m + 4 2 m + 4 1 ( 2 n ) ! m + 1 n + m + 1 ∣ B 2 n + 2 m + 2 ∣ ∣ B 2 m + 2 ∣ 1 2 n + 2 m + 2 2 m + 2 1 ( 2 n ) ! = ( 2 m + 3 ) ( m + 2 ) 2 m + 1 B 2 m + 2 B 2 m + 4 × n + m + 1 ( n + m + 2 ) 2 ( 2 n + 2 m + 3 ) B 2 n + 2 m + 4 B 2 n + 2 m + 2

for m ∈ N and n ∈ N 0 . By virtue of Lemma 2, we see that the sequence (8) is increasing in n ∈ N 0 for given m ∈ N . Further with the help of Lemma 1, we derive the conclusion that the ratio H m + 1 ( t ) H m ( t ) is increasing in t ∈ ( 0 , 2 π ) for given m ∈ N . Considering the evenness of H m ( t ) defined by (3), we complete the proof of the monotonicity of the ratio H m + 1 ( t ) H m ( t ) stated in Theorem 3.□

6 Connections with generalized hypergeometric functions

In terms of the rising factorial, also known as the Pochhammer symbol,

( ϑ ) n = ∏ ℓ = 0 n − 1 ( ϑ + ℓ ) = ϑ ( ϑ + 1 ) … ( ϑ + n − 1 ) , n ∈ N ; 1 , n = 0

for α 1 ∈ C and β 1 , β 2 ∈ C \ { 0 , − 1 , − 2 , … } , the generalized hypergeometric function F 2 1 ( α 1 ; β 1 , β 2 ; z ) is defined [5, p. 124] by

F 2 1 ( α 1 ; β 1 , β 2 ; z ) = ∑ n = 0 ∞ ( α 1 ) n ( β 1 ) n ( β 2 ) n z n n ! , z ∈ C ;

see [16, Chapter II], [17, p. 1020], and [18, Chapter 14].

As done in [2, Section 5] and [9, Remark 7], by virtue of the relation

CosR n ( t ) = F 2 1 1 ; n + 1 2 , n + 1 ; − t 2 4 , n ∈ N

reformulated in [4, p. 16], we can write some main results in this article as follows:

For m ∈ N , the function
− 1 t 2 m + 2 ln CosR 1 ( t ) + ∑ n = 1 m ∣ B 2 n ∣ n t 2 n ( 2 n ) ! = − 1 t 2 m + 2 ln F 2 1 1 ; 3 2 , 2 ; − t 2 4 + ∑ n = 1 m ∣ B 2 n ∣ n t 2 n ( 2 n ) !
is logarithmically convex in t ∈ ( − 2 π , 2 π ) \ { 0 } .
For m ∈ N , the function
1 t 2 + ∣ B 2 m + 2 ∣ ( m + 1 ) ( 2 m + 2 ) ! t 2 m ln F 2 1 1 ; 3 2 , 2 ; − t 2 4 + ∑ n = 1 m ∣ B 2 n ∣ n t 2 n ( 2 n ) !
is increasing in t ∈ ( 0 , 2 π ) and decreasing in t ∈ ( − 2 π , 0 ) .

7 Conclusions

The main conclusions in this article include three theorems: Theorems 1–3.

Theorem 1 read that Qi’s normalized remainder H m ( t ) for m ∈ N is logarithmically convex in t ∈ ( − 2 π , 2 π ) .

Theorem 2 stated that the logarithm of Qi’s normalized remainder H m ( t ) for m ∈ N can be expanded into the Maclaurin series

ln H m ( t ) = − ∑ ℓ = 1 ∞ M m , 2 ℓ ω 2 ( m + 1 ) 2 ℓ t 2 ℓ ( 2 ℓ ) ! , ∣ t ∣ < 2 π .

Theorem 3 wrote that the ratio H m + 1 ( t ) H m ( t ) for m ∈ N is increasing in t ∈ ( 0 , 2 π ) and is decreasing in t ∈ ( − 2 π , 0 ) .

The concept of normalized remainders (also known as normalized tails) of the Maclaurin series expansions of infinitely differentiable functions, created and designed by Feng Qi step by step and little by little since April 2023, is new, novel, and significant in mathematics. About the idea and thought to initially invent the concept of normalized remainders, please read [6, Section 5], [7, Section 1], and [19, Section 1]. In [1–4,6–9,19–21], the monotonicity, the convexity, the Maclaurin series expansions of the logarithms of the normalized remainders, and other properties for the elementary functions cos z , tan z , sin z , tan 2 z , e z , and z e z − 1 were, respectively, considered by close ideas and tools.

Acknowledgements

The authors are grateful to anonymous referees for their careful reading, valuable suggestions, helpful comments on the original version of this article.

Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
Conflict of interest: The authors state no conflicts of interest.
Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

[1] Y.-F. Li and F. Qi, A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine, Open Math. 21 (2023), no. 1, 20230159, DOI: https://doi.org/10.1515/math-2023-0159. 10.1515/math-2023-0159Search in Google Scholar

[2] D.-W. Niu and F. Qi, Monotonicity results of ratios between normalized tails of Maclaurin power series expansions of sine and cosine, Mathematics 12 (2024), no. 12, 1781, DOI: https://doi.org/10.3390/math12121781. 10.3390/math12121781Search in Google Scholar

[3] A. Wan and F. Qi, Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine, Electron. Res. Arch. 32 (2024), no. 5, 3130–3144, DOI: https://doi.org/10.3934/era.2024143. 10.3934/era.2024143Search in Google Scholar

[4] T. Zhang, Z.-H. Yang, F. Qi, and W.-S. Du, Some properties of normalized tails of Maclaurin power series expansions of sine and cosine, Fractal Fract. 8 (2024), no. 5, 257, DOI: https://doi.org/10.3390/fractalfract8050257. 10.3390/fractalfract8050257Search in Google Scholar

[5] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, DOI: http://doi.org/10.1002/9781118032572. 10.1002/9781118032572Search in Google Scholar

[6] Z.-H. Bao, R. P. Agarwal, F. Qi, and W.-S. Du, Some properties on normalized tails of Maclaurin power series expansion of exponential function, Symmetry 16 (2024), no. 8, 989, DOI: https://doi.org/10.3390/sym16080989. 10.3390/sym16080989Search in Google Scholar

[7] G.-Z. Zhang and F. Qi, On convexity and power series expansion for logarithm of normalized tail of power series expansion for square of tangent, J. Math. Inequal. 18 (2024), no. 3, 937–952, DOI: https://doi.org/10.7153/jmi-2024-18-51. 10.7153/jmi-2024-18-51Search in Google Scholar

[8] G.-Z. Zhang, Z.-H. Yang, and F. Qi, On normalized tails of series expansion of generating function of Bernoulli numbers, Proc. Amer. Math. Soc. 153 (2025), no. 1, in press, DOI: https://doi.org/10.1090/proc/16877.10.1090/proc/16877Search in Google Scholar

[9] Y.-W. Li and F. Qi, A new closed-form formula of the Gauss hypergeometric function at specific arguments, Axioms 13 (2024), no. 5, 317, DOI: https://doi.org/10.3390/axioms13050317. 10.3390/axioms13050317Search in Google Scholar

[10] M. Biernacki and J. Krzyż, On the Monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 9 (1955), 135–147. Search in Google Scholar

[11] S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), no. 2, 278–301, DOI: https://doi.org/10.1112/S0025579300012602. 10.1112/S0025579300012602Search in Google Scholar

[12] Z.-H. Yang, Y.-M. Chu, and M.-K. Wang, Monotonicity criterion for the quotient of power series with applications, J. Math. Anal. Appl. 428 (2015), no. 1, 587–604, DOI: https://doi.org/10.1016/j.jmaa.2015.03.043. 10.1016/j.jmaa.2015.03.043Search in Google Scholar

[13] Y. Shuang, B.-N. Guo, and F. Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 3, 135, DOI: https://doi.org/10.1007/s13398-021-01071-x. 10.1007/s13398-021-01071-xSearch in Google Scholar

[14] P. Cerone and S. S. Dragomir, Some convexity properties of Dirichlet series with positive terms, Math. Nachr. 282 (2009), no. 7, 964–975, DOI: https://doi.org/10.1002/mana.200610783. 10.1002/mana.200610783Search in Google Scholar

[15] N. Bourbaki, Elements of Mathematics: Functions of a Real Variable: Elementary Theory, Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004, DOI: https://doi.org/10.1007/978-3-642-59315-4. 10.1007/978-3-642-59315-4Search in Google Scholar

[16] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. I, Based on notes left by Harry Bateman, with a preface by Mina Rees, with a foreword by E. C. Watson. Reprint of the 1953 original, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1981. Search in Google Scholar

[17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015, DOI: https://doi.org/10.1016/B978-0-12-384933-5.00013-8. 10.1016/B978-0-12-384933-5.00013-8Search in Google Scholar

[18] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis–An Introduction to the General Theory of Infinite Processes and of Analytic Functions with an Account of the Principal Transcendental Functions, Fifth edition, Edited by Victor H. Moll, with a foreword by S. J. Patterson, Cambridge University Press, Cambridge, 2021. Search in Google Scholar

[19] F. Qi, Absolute monotonicity of normalized tail of power series expansion of exponential function, Mathematics 12 (2024), no. 18, 2859, DOI: https://doi.org/10.3390/math12182859. 10.3390/math12182859Search in Google Scholar

[20] Y.-W. Li, F. Qi, and W.-S. Du, Two forms for Maclaurin power series expansion of logarithmic expression involving tangent function, Symmetry 15 (2023), no. 9, 1686, DOI: https://doi.org/10.3390/sym15091686. 10.3390/sym15091686Search in Google Scholar

[21] X.-L. Liu, H.-X. Long, and F. Qi, A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing sine, Mathematics 11 (2023), no. 14, 3107, DOI: https://doi.org/10.3390/math11143107. 10.3390/math11143107Search in Google Scholar

Received: 2024-09-17

Revised: 2024-11-01

Accepted: 2024-11-07

Published Online: 2024-12-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0095

Keywords for this article

Maclaurin series expansion; Qi’s normalized remainder; Bernoulli number; monotonicity; logarithmic convexity; monotonicity rule; derivative formula; cosine; ratio

Creative Commons

BY 4.0