Qi's Normalized Remainders

• Qi’s normalized remainders are Taylor series truncation errors scaled by the first nonzero term, producing a normalized value of 1 at the expansion point.
• They enable precise analysis of properties such as monotonicity, convexity, and absolute monotonicity across elementary functions like exponential, sine, cosine, and tangent.
• The framework unifies analytic techniques with combinatorial generating functions to yield new insights in asymptotic expansion, number theory, and related fields.

Searching arXiv for the cited works and recent relevant sources on normalized remainders. Querying arXiv for "normalized remainders Qi Maclaurin Taylor". Qi’s normalized remainders are normalized forms of Taylor or Maclaurin truncation errors in which the remainder after removing the first $n+1$ terms is divided by its first nonzero local model, producing a dimensionless function equal to $1$ at the expansion point. In the notation surveyed by Qi, they are written $T_n[f(x),x_0]$, with the Maclaurin shorthand $T_n[f(x)]$, and they are studied as analytic objects in their own right rather than merely as error terms in approximation formulas (Qi, 16 Dec 2025).

1. Definition and normalization principle

Let

$$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$

be the partial sum of the expansion of a smooth function $f$ around an interior point $x_0$. The ordinary Taylor remainder is

$R_n(x)=f(x)-S_n(x,x_0).$

Qi’s framework normalizes this remainder by its first nonzero local factor

$\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$

so that the normalized remainder satisfies $T_n[f(x_0),x_0]=1$ (Qi, 16 Dec 2025).

The same chapter also includes the case

$1$0

when $1$1. For $1$2, the existence assumptions are that $1$3 is infinitely differentiable on an interval $1$4, $1$5 is an interior point of $1$6, and $1$7 (Qi, 16 Dec 2025).

This normalization removes the trivial vanishing of the raw remainder at $1$8. The result is a finite object with a fixed anchor value $1$9, which can then be investigated for monotonicity, convexity, logarithmic convexity or concavity, absolute monotonicity, complete monotonicity, integral representations, hypergeometric forms, and generating-function interpretations (Qi, 16 Dec 2025).

Several low-order examples are canonical. The survey records

$T_n[f(x),x_0]$0

together with

$T_n[f(x),x_0]$1

These illustrate the basic principle: a familiar quotient is reinterpreted as a normalized tail of a power series (Qi, 16 Dec 2025).

The chapter also lists basic invariance properties. If $T_n[f(x),x_0]$2 exists, then

$T_n[f(x),x_0]$3

for $T_n[f(x),x_0]$4, and

$T_n[f(x),x_0]$5

for $T_n[f(x),x_0]$6. If $T_n[f(x),x_0]$7 for $T_n[f(x),x_0]$8 and $T_n[f(x),x_0]$9, then

$T_n[f(x)]$0

These formulas make the normalization robust under the simplest affine operations and under extraction of known vanishing factors (Qi, 16 Dec 2025).

2. Origins, terminology, and historical background

The survey states that the terminology normalized remainder first appeared explicitly in 2023 and also records the alternative label normalized tail; some later papers call it Qi’s normalized remainder (Qi, 16 Dec 2025). At the same time, the survey argues that the underlying structure is older and had long been present implicitly in combinatorics and number theory.

The explicit analytic origin is traced to early studies of tangent, sine, and cosine. The functions

$T_n[f(x)]$1

were recognized as normalized remainders $T_n[f(x)]$2, $T_n[f(x)]$3, $T_n[f(x)]$4, $T_n[f(x)]$5, and $T_n[f(x)]$6, respectively. This recognition led to the systematic study of the functions $T_n[f(x)]$7 as independent analytic objects (Qi, 16 Dec 2025).

The chapter’s historical thesis is broader. Classical generating functions for Bernoulli numbers, Bernoulli polynomials, generalized Bernoulli polynomials, Stirling numbers, Stirling polynomials, and their extensions by Howard, Carlitz, and Broder can all be rewritten through normalized remainders. The generating function

$T_n[f(x)]$8

is identified with

$T_n[f(x)]$9

and

$$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$0

with

$$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$1

This indicates that the explicit concept is recent, while many of its algebraic avatars are not (Qi, 16 Dec 2025).

A recurring theme is therefore conceptual rather than merely terminological: normalized remainders turn local truncation errors into global functions whose positivity, growth, and coefficient behavior can be examined directly. This suggests why the notion moved quickly from elementary-function inequalities to generating functions and then to broader analogical settings.

3. Elementary-function families and basic properties

The survey reviews a substantial body of recent results for elementary functions, especially since 2023. The most developed cases concern exponential, trigonometric, inverse trigonometric, Bernoulli-generating, and related special-function families (Qi, 16 Dec 2025).

Family	Representative form	Surveyed properties
Exponential	$$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$2	positive, increasing, logarithmically convex, absolutely monotonic
Sine	$$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$3	positive and decreasing on $$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$4; concave on $$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$5 for $$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$6
Cosine	$$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$7	positive and decreasing on $$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$8; concave on $$S_n(x,x_0)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,\qquad n\in\mathbb N_0,$$9 for $f$0
Tangent	$f$1	even; absolutely monotonic on $f$2; completely monotonic on $f$3
Inverse trigonometric	$f$4, $f$5	positive and even on $f$6; monotonicity depends on the function

For the exponential family, the integral representation

$f$7

is central. It supports the statements that $f$8 is positive on $f$9, increasing on $x_0$0, logarithmically convex on $x_0$1, and absolutely monotonic on $x_0$2. The same body of work also shows that

$x_0$3

is decreasing on $x_0$4, and the survey records the guess that $x_0$5 is logarithmically absolutely monotonic on $x_0$6 (Qi, 16 Dec 2025).

For sine and cosine, the survey gives both integral and hypergeometric representations: $x_0$7

$x_0$8

These formulas are used to derive positivity, monotonicity, concavity, and ratio inequalities between adjacent normalized remainders (Qi, 16 Dec 2025).

The tangent family occupies a special place in the origin story. For $x_0$9, the survey states that $R_n(x)=f(x)-S_n(x,x_0).$0 is even on $R_n(x)=f(x)-S_n(x,x_0).$1, absolutely monotonic on $R_n(x)=f(x)-S_n(x,x_0).$2, completely monotonic on $R_n(x)=f(x)-S_n(x,x_0).$3, logarithmically absolutely monotonic on $R_n(x)=f(x)-S_n(x,x_0).$4, and logarithmically completely monotonic on $R_n(x)=f(x)-S_n(x,x_0).$5. It also records monotonicity and concavity results for

$R_n(x)=f(x)-S_n(x,x_0).$6

These are among the strongest order-theoretic statements currently attached to a normalized-remainder family (Qi, 16 Dec 2025).

The survey extends the program to $R_n(x)=f(x)-S_n(x,x_0).$7, $R_n(x)=f(x)-S_n(x,x_0).$8, $R_n(x)=f(x)-S_n(x,x_0).$9, $\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$0, the reciprocal-gamma-integral function

$\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$1

and powers $\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$2. In the last case it records that, for $\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$3 and $\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$4,

$\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$5

is positive, increasing, convex, absolutely monotonic, and logarithmically convex on $\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$6, and that the coefficients $\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$7 in the expansion of $\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$8 satisfy

$\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1},$9

linking the subject to central factorial numbers of the first kind (Qi, 16 Dec 2025).

4. Threshold phenomena for the degenerate exponential

A 2026 paper studies a degenerate analogue of Qi’s normalized remainder for the degenerate exponential

$T_n[f(x_0),x_0]=1$0

defined on

$T_n[f(x_0),x_0]=1$1

The paper defines a normalized remainder $T_n[f(x_0),x_0]=1$2 and shows that, in the limit $T_n[f(x_0),x_0]=1$3, it recovers the classical normalized remainder

$T_n[f(x_0),x_0]=1$4

for the ordinary exponential (Suna et al., 30 Jun 2026).

The foundational formula is the integral representation

$T_n[f(x_0),x_0]=1$5

From this, the paper derives exact derivative formulas and an exact monotonicity threshold at

$T_n[f(x_0),x_0]=1$6

More precisely, $T_n[f(x_0),x_0]=1$7 is strictly increasing on $T_n[f(x_0),x_0]=1$8 if $T_n[f(x_0),x_0]=1$9, identically $1$00 if $1$01, and strictly decreasing if $1$02 (Suna et al., 30 Jun 2026).

The same paper shows that the classical exponential picture does not carry over unchanged. Writing

$1$03

it proves global strict logarithmic concavity on $1$04 when

$1$05

and establishes local failure of logarithmic convexity at the origin when

$1$06

The most decisive result is asymptotic: $1$07 for every $1$08. Hence global logarithmic convexity on $1$09 fails throughout the entire increasing regime (Suna et al., 30 Jun 2026).

Absolute monotonicity also becomes exceptional rather than generic. The paper proves that $1$10 is absolutely monotonic on $1$11 if and only if

$1$12

equivalently

$1$13

This admissible set is countably infinite, discrete in $1$14, has only accumulation point at $1$15, and is therefore of Lebesgue measure zero (Suna et al., 30 Jun 2026).

The paper also gives explicit two-sided truncation-error bounds. In the increasing regime, for $1$16,

$1$17

with corresponding explicit bounds for the raw truncation error $1$18. These bounds are pointwise sharp at the origin (Suna et al., 30 Jun 2026).

5. Combinatorial and generating-function reformulations

One of the clearest uses of Qi’s normalized remainders is as a common analytic language for classical and generalized combinatorial generating functions. A 2025 note emphasizes that the notion is not used there to prove new intrinsic theorems about normalized remainders; instead, it is used to reformulate the definitions of Stirling numbers and their extensions by Howard, Carlitz, and Broder (He et al., 11 Sep 2025).

At the most basic level,

$1$19

These two zeroth normalized remainders already encode the classical Stirling numbers. The note rewrites the standard generating functions as

$1$20

and

$1$21

so the ordinary Stirling numbers of the second and first kinds become coefficient arrays of powers of normalized remainders (He et al., 11 Sep 2025).

Higher normalized remainders generate associated Stirling families. For Howard’s associated Stirling numbers of the second kind, the note records

$1$22

For Howard’s associated Stirling numbers of the first kind it gives the corresponding logarithmic formulation in terms of $1$23. Broder’s $1$24-Stirling numbers of the first kind appear through

$1$25

while Broder’s and Carlitz’s $1$26-Stirling numbers of the second kind appear through

$1$27

The normalized-remainder language thereby places multiple coefficient families into a single analytic template (He et al., 11 Sep 2025).

The broader survey chapter makes the same point historically. It rewrites Bernoulli-number and Bernoulli-polynomial generating functions through $1$28, and it rewrites Howard–Carlitz generating functions such as

$1$29

as

$1$30

This suggests that normalized remainders function as a unifying analytic layer over older combinatorial constructions (Qi, 16 Dec 2025).

A common misconception is that these combinatorial papers establish new monotonicity or convexity laws for $1$31 themselves. The 2025 note explicitly does not do so: its contribution is organizational and reformulatory, not foundational, and its use of normalized remainders is primarily expository and conceptual (He et al., 11 Sep 2025).

6. Asymptotic remainders, arithmetic analogues, and changes of context

The notion of normalization also appears in asymptotic analysis of special functions. In work motivated by conjectures of Qi and Liu, Qi and Mahmoud study the remainders

$1$32

for the asymptotic expansion of $1$33, together with the differentiated remainders

$1$34

Here the normalization is by powers of $1$35, and the central invariant is the completely monotonic degree $1$36, defined as the largest exponent $1$37 such that $1$38 remains completely monotonic (Qi et al., 2019).

The paper proves

$1$39

and

$1$40

For $1$41, it obtains the two-sided bound

$1$42

Its key Laplace-transform representation is

$1$43

where

$1$44

This is a distinct normalization regime from the Maclaurin-tail setting, but it preserves the central idea of extracting the scale at which a remainder exhibits structured positivity (Qi et al., 2019).

Outside analysis in the narrow sense, several papers use a “Qi-style” or normalized-remainder lens without adopting the formal definition. In robust generalized CRT for two integers, the decomposition

$1$45

separates each unknown into a common remainder $1$46 and a quotient or folding part $1$47. The paper interprets the reduced values

$1$48

as noisy normalized or common remainders and builds a reconstruction algorithm around clustering them modulo $1$49, with robustness guaranteed when the remainder error satisfies

$1$50

and final error bounded by $1$51 (Li et al., 2015).

In norm-Euclidean real quadratic fields, an explicit division algorithm is constructed for

$1$52

and the paper can be read through the normalized-remainder viewpoint by setting

$1$53

The field-specific coverings by hyperbolic strips guarantee

$1$54

hence

$1$55

The paper does not use the phrase “Qi’s normalized remainders,” but it develops an explicit quotient-selection mechanism from which such normalized remainders may be derived (Morain, 6 Feb 2026).

A further analogue appears in the Euclidean algorithm with symmetric quotients. Smith derives exact formulas for paired remainders

$1$56

$1$57

with

$1$58

Then many normalized quadratic combinations,

$1$59

are themselves remainders from shorter symmetric Euclidean algorithms. This is not the Maclaurin theory, but it exhibits the same principle of dividing a remainder expression by its natural scale to reveal recursive structure (Smith, 2013).

These extensions suggest a distinction that is often obscured in informal usage. In the strict sense, Qi’s normalized remainders are Maclaurin or Taylor normalized tails. In broader current usage, the phrase may designate any setting where a remainder-like quantity is divided by its first natural scale or anchor to expose monotonicity, convexity, reconstructibility, or algebraic recursion.

7. Open problems, current scope, and interpretive cautions

The survey chapter presents normalized remainders as an emerging framework rather than a closed theory. It records explicit problems on expanding

$1$60

and on studying monotonicity and convexity of ratios such as

$1$61

It also proposes investigation of generalized coefficient arrays generated by

$1$62

and

$1$63

which are intended to unify Bernoulli, Howard, and Stirling-type structures (Qi, 16 Dec 2025).

The chapter additionally records several guesses: that $1$64 is logarithmically absolutely monotonic on $1$65, that

$1$66

is absolutely monotonic on $1$67, that certain tangent-square ratios are concave and even absolutely or completely monotonic on their natural intervals, and that various inverse-trigonometric adjacent-ratio functions exhibit stronger shape properties than currently proved (Qi, 16 Dec 2025).

Two interpretive cautions are important. First, the terminology is recent, but the structures are older. The survey’s historical claim is not that normalized remainders suddenly created Bernoulli or Stirling theory, but that they provide a new analytic lens through which these older constructions can be read (Qi, 16 Dec 2025). Second, not every paper using normalized-remainder language is about the same object. The combinatorial note on Stirling numbers uses the concept mainly as a unifying language, not as the subject of new intrinsic theorems; the robust CRT, Euclidean-algorithm, and real-quadratic-field papers invoke normalized-remainder viewpoints only by analogy or reinterpretation (He et al., 11 Sep 2025).

Taken together, the current literature supports a coherent but multi-layered picture. At its core, Qi’s normalized remainder is the normalized tail of a Taylor or Maclaurin series, anchored at value $1$68 and studied for global analytic structure. Around that core has grown a broader program connecting inequalities, complete and absolute monotonicity, integral and hypergeometric representations, combinatorial generating functions, asymptotic expansions, and several arithmetic analogies. This suggests that the concept’s long-term significance may lie less in a single formula than in a normalization principle that repeatedly turns local remainder terms into globally structured functions (Qi, 16 Dec 2025).