Some special functions and cylindrical diffusion equation on α-time scale

Burcu Silindir; Zehra Tuncer; Seçil Gergün; Ahmet Yantir

doi:10.1515/dema-2025-0131

Article Open Access

Some special functions and cylindrical diffusion equation on α-time scale

Burcu Silindir , Zehra Tuncer , Seçil Gergün and Ahmet Yantir

Published/Copyright: June 10, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This article is dedicated to present various concepts on α -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the α -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the α -Gauss binomial formula. Furthermore, we define the α -gamma function and prove α -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the α -gamma function is the unique logarithmically convex solution of f ( s + 1 ) = ϕ ( s ) f ( s ) , f ( 1 ) = 1 , where ϕ ( s ) refers to the α -number. In addition, we present Euler’s infinite product form and asymptotic behavior of α -gamma function. As an application, we propose α -analogue of the cylindrical diffusion equation, from which α -Bessel and modified α -Bessel equations are derived. We explore the solutions of the α -cylindrical diffusion equation using the separation of variables technique, revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally, we illustrate the graphs of the α -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.

Keywords: exponential function; gamma function; Bessel equations; Bessel functions; cylindrical diffusion equation

MSC 2010: 26E70; 34N05; 33D05; 33C10; 33B15; 34B30

1 Introduction

In [1], we introduced the weighted operator α

(1) α ( x ) ≔ t x − h q + ( 1 − t ) ( q x + h ) , x ∈ R , h ∈ R 0 + , q ∈ R + , t ∈ [ 0 , 1 ] ,

determined by the convex sum of the backward and forward jump operators of the ( q , h ) -time scale [2]. The α -operator is a generic operator, which leads to unification for t ∈ { 0 , 1 } and extension for t ∈ ( 0 , 1 ) . If, for example, t = q 4 ⁄ 3 − q 2 1 − q 2 and 0 < q < 1 , then t ∈ ( 0 , 1 ) , and in this case, α -operator plays the following extended role

α ( x ) = q 1 ⁄ 3 x + h 1 − q 1 ⁄ 3 1 − q ,

leading to “1/3 jump action” as α ( x ) = q 1 ⁄ 3 x for h = 0 and α ( x ) = x + h 3 as q → 1 . Generated by the α -operator, for x 0 ≠ h 1 − q , we introduced the α -time scale [1] by

(2) T α x 0 ≔ { α n ( x 0 ) : n ∈ Z } ∪ h 1 − q .

The time scale theory is an innovative approach that integrates continuous and discrete analysis, facilitating the exploration of dynamical systems in a more comprehensive manner [3–5]. However, the polynomials can be represented only implicitly and recursively through Δ -polynomials [6,7], ∇ -polynomials [8], and diamond-alpha polynomials [9]. Therefore, the elementary concepts like polynomials, power functions, Taylor series, or exponential functions remain undeveloped in clear, efficient, and applicable forms within a general time scale context.

The study on α -time scale is a novel framework that unifies and extends the studies on nabla ( q , h ) -, delta ( q , h ) -, nabla h -, delta h -, nabla q -, and delta q -analyses. At least as importantly, it allows us to construct the fundamental functions such as the polynomials [1], and the power functions that are explicitly determined in terms of α -operator and the logarithm function [10] whose construction is inspired by the linearly independent solutions of an α -Cauchy-Euler equation. Since the current article is a continuation paper, Section 2 is reserved for the list of the main definitions, some important results of [1] and [10], and some new tools to explain the nature of α -time scale and its analysis.

The current article is dedicated to develop the α -analogues of Taylor series, exponential function, gamma function, and Bessel functions of the first kind. To achieve this, in Section 3, we introduce α -analogue of power series, the conditions for its absolute and uniform convergence as well as its term-by-term α -differentiation and develop α -analogue of Taylor series equipped with the notion of analyticity. In addition, we present α -binomial series and prove its α -analyticity by showing that the same IVP admits both the power function and its Taylor series as the unique solution. Section 4 explores the α -exponential function as a series and the criteria for its absolute and uniform convergence. Furthermore, by the use of α -analogue of Gauss’s binomial formula, we state and prove the additive identity of α -exponential function. Such additive identity is the core of not only discovering the form of the α -gamma function but also the main theorem stating and proving that the IVP, f ( s + 1 ) = ϕ ( s ) f ( s ) , f ( 1 ) = 1 , is satisfied by the α -gamma function, where ϕ ( s ) is the α -number. We also prove that α -gamma function is logarithmically convex; moreover, α -gamma function is the unique function that meets the requirements of the Bohr-Mollerup theorem. In addition, we present Euler’s infinite product form and asymptotic behavior of α -gamma function. Section 5 addresses an α -partial difference equation, which not only admits solutions including α -exponential and α -gamma functions but also enables us to explore α -analogues of Bessel and modified Bessel functions. To achieve this, we offer α -analogue of cylindrical diffusion equation whose coefficient functions are determined in terms of α -polynomials. By using the separation of variables method, we obtain two α -ordinary difference equations. We examine the solutions of α -cylindrical diffusion equation for different values of the constant η . For η = 0 , the solution is in logarithmic form. For η > 0 , one of the α -ordinary difference equation has solution in terms of the α -exponential function, while the second α -ordinary difference equation turns out to be α -analogue of Bessel equation of order zero. We construct α -Bessel function of the first kind. Similar to the ordinary case, α -Bessel function is characterized by the α -gamma function. In a similar fashion, for η < 0 , we encounter modified α -Bessel equation of order zero and establish modified α -Bessel function of the first kind. We present the reductions of α -exponential and α -gamma functions to their discrete and ordinary counterparts. We also illustrate α -exponential and α -gamma functions for several q , h , t , k values. The historical and mathematical importance of the functions under consideration, coupled with the innovative approach presented through the α -time scale, underscores the value of this study in advancing the theoretical understanding and practical applications of these fundamental mathematical concepts.

2 Fundamentals of α -time scale

We introduced the concept of α -time scale in [1] and presented its analysis in [10]. We devote this section to collect main definitions and some beneficial results of [1,10] about the calculus on α -time scale in addition to some new properties that will be used throughout the current article.

To explain the nature of α -time scale, we present the following proposition [10], which lists some core properties of the α -operator and the α k -number, which is defined as follows:

(3) [ r ] α k ≔ t q + ( 1 − t ) q r k − 1 t q + ( 1 − t ) q k − 1 if t q + ( 1 − t ) q ≠ 1 , r if t q + ( 1 − t ) q = 1 ,

for t ∈ [ 0 , 1 ] and q ∈ R + .

Proposition 2.1

[ 0 ] α k = 0 and [ 1 ] α k = 1 .
If t = q q + 1 or q = 1 , then t q + ( 1 − t ) q = 1 and hence [ r ] α k = r for all r ∈ R .
The α k -numbers have the following limits:
lim r → ∞ [ r ] α k = 1 1 − t q + ( 1 − t ) q k if 0 < t q + ( 1 − t ) q k < 1 , ∞ if 1 ≤ t q + ( 1 − t ) q k ,
and
lim r → − ∞ [ r ] α k = − ∞ if 0 < t q + ( 1 − t ) q k ≤ 1 , 1 1 − t q + ( 1 − t ) q k if 1 < t q + ( 1 − t ) q k .
For r ∈ R , we have
α r ( x ) = t q + ( 1 − t ) q r x − h 1 − q + h 1 − q .
For r ∈ R , α r ( x ) = x iff r = 0 or x = h 1 − q or t q + ( 1 − t ) q = 1 .
The following limits hold for α r -operator
lim r → ∞ α r ( x ) = ∞ if t q + ( 1 − t ) q > 1 and x > h 1 − q , − ∞ if t q + ( 1 − t ) q > 1 and x < h 1 − q , h 1 − q if t q + ( 1 − t ) q < 1 ,
and
lim r → − ∞ α r ( x ) = h 1 − q if t q + ( 1 − t ) q > 1 , ∞ if t q + ( 1 − t ) q < 1 and x > h 1 − q , − ∞ if t q + ( 1 − t ) q < 1 and x < h 1 − q .
For all ℓ , r , s ∈ R , we have
α s ( x ) − α r ( a ) = t q + ( 1 − t ) q ℓ ( α s − ℓ ( x ) − α r − ℓ ( a ) ) .

Proposition 2.1 (vi) assures that the point h 1 − q is the accumulation point of α -time scale (2). Moreover, by Proposition 2.1 (iv), x > h 1 − q if and only if α r ( x ) > h 1 − q , and therefore, the elements of T α x 0 are either located to the right of h 1 − q or to the left of h 1 − q depending on whether x 0 > h 1 − q or x 0 < h 1 − q . For simplicity, in what follows, we assume that the seed of the time scale x 0 > h 1 − q , and hence, for all x ∈ T α x 0 , r ∈ R , we have α r ( x ) ≥ h 1 − q and any function under consideration is real valued and defined for x ≥ h 1 − q . In [1], for any k ∈ Z , we defined the α k -derivative of f by

(4) D α k f ( x ) ≔ f ( α k ( x ) ) − f ( x ) α k ( x ) − x if x ≠ h 1 − q , lim s → h 1 − q s ∈ T α x 0 f ( s ) − f h 1 − q s − h 1 − q = f ′ h 1 − q if x = h 1 − q ,

if the limit exists. Note that, the α k -derivative (4) is generic. If t is 0 or 1, it provides nabla and delta ( q , h ) -derivatives [11,12], and nabla and delta q -derivatives [13], and nabla and delta h -derivatives [3]. In addition, if t ∈ ( 0 , 1 ) , α k -derivative extends the ( q , h ) -derivatives as well as q - and h -derivatives. For details, see [1].

In [1], we presented the product rule

(5) D α k ( f ( x ) g ( x ) ) = f ( α k ( x ) ) D α k g ( x ) + g ( x ) D α k f ( x ) = g ( α k ( x ) ) D α k f ( x ) + f ( x ) D α k g ( x ) ,

and the quotient rule

(6) D α k f ( x ) g ( x ) = g ( x ) D α k f ( x ) − f ( x ) D α k g ( x ) g ( x ) g ( α k ( x ) ) .

Moreover, for a , b , c > h 1 − q with a < b , we defined the α k -definite integrals [10] of f by

∫ h 1 − q , c f ( x ) d α k x ≔ 1 − t q + ( 1 − t ) q k c − h 1 − q ∑ j = 0 ∞ t q + ( 1 − t ) q k j f ( α k j ( c ) )

and

∫ ( a , b ] f ( x ) d α k x ≔ ∫ h 1 − q , b f ( x ) d α k x − ∫ h 1 − q , a f ( x ) d α k x

whenever the series converge. If F is an α k -antiderivative of f that is continuous at h 1 − q and h 1 − q ≤ a < b , then

(7) ∫ ( a , b ] f ( x ) d α k x = F ( b ) − F ( a ) .

Specifically, if f is continuous at h 1 − q and h 1 − q ≤ a < b , then

(8) ∫ ( a , b ] D α k f ( x ) d α k x = f ( b ) − f ( a ) .

For details see [10]. Now, we state the α k -integration by parts formula, which will be utilized for constructing the α -gamma function.

Theorem 2.2

If f and g are continuous at h 1 − q , then

∫ ( a , b ] f ( x ) D α k g ( x ) d α k x = f ( b ) g ( b ) − f ( a ) g ( a ) − ∫ ( a , b ] g ( α k ( x ) ) D α k f ( x ) d α k x .

Proof

It follows from (8) and the product rule (5) that

f ( b ) g ( b ) − f ( a ) g ( a ) = ∫ ( a , b ] D α k ( f ( x ) g ( x ) ) d α k x = ∫ ( a , b ] g ( α k ( x ) ) D α k f ( x ) d α k x + ∫ ( a , b ] f ( x ) D α k g ( x ) d α k x ,

which finishes the proof.□

In [1], for k ∈ Z , m , n ∈ N 0 , we defined the α k -polynomials as follows:

(9) ( x − a ) α k m = ( x − a ) ( x − α k ( a ) ) ( x − α 2 k ( a ) ) … ( x − α ( m − 1 ) k ( a ) ) = ∏ j = 0 m − 1 ( x − α j k ( a ) ) ,

proved that it has the additivity property

( x − a ) α k m + n = ( x − a ) α k m ( x − α m k ( a ) ) α k n

and has the power rule for derivative

(10) D α k ( x − a ) α k m = [ m ] α k ( x − a ) α k m − 1 .

Note that when m ≠ 0 , 1 ,

( a − x ) α k m ≠ ( − 1 ) m ( x − a ) α k m .

Instead, we have the relation in Proposition 2.3.

Proposition 2.3

For any k ∈ Z , m ∈ N 0 ,

( a − x ) α k m = ( − 1 ) m t q + ( 1 − t ) q m ( m − 1 ) k 2 ( x − a ) α − k m .

Proof

The formula is obvious when m = 0 . If m ∈ N , by Proposition 2.1 (vii), we have

□ ( a − x ) α k m = ( a − x ) ( a − α k ( x ) ) ( a − α 2 k ( x ) ) … ( a − α ( m − 1 ) k ( x ) ) = ( − 1 ) m ( x − a ) ( α k ( x ) − a ) ( α 2 k ( x ) − a ) … ( α ( m − 1 ) k ( x ) − a ) = ( − 1 ) m t q + ( 1 − t ) q m ( m − 1 ) k 2 ( x − a ) ( x − α − k ( a ) ) ( x − α − 2 k ( a ) ) … ( x − α − ( m − 1 ) k ( a ) ) = ( − 1 ) m t q + ( 1 − t ) q m ( m − 1 ) k 2 ( x − a ) α − k m .

In [10], we introduced an α k -power function, denoted by ( a + x ) α k r . To relate with the polynomial (9), we offer another well-defined power function introduced as follows.

Definition 2.4

For k ∈ Z , a > h 1 − q , r ∈ R , h 1 − q ≤ α k r ( x ) < a , and t q + ( 1 − t ) q k < 1 , we define the α k -power function, denoted by ( a − x ) α k r as follows:

(11) ( a − x ) α k r ≔ a − h 1 − q r ∏ j = 0 ∞ a − α k j ( x ) a − α k ( r + j ) ( x ) .

For any r ∈ R , α r h 1 − q = h 1 − q , and hence, a − h 1 − q α k r = a − h 1 − q r .

Proposition 2.5

The infinite product (11) converges absolutely for t q + ( 1 − t ) q k < 1 and diverges for t q + ( 1 − t ) q k > 1 .

Proof

When α k r ( x ) < a , we have

α k ( r + j ) ( x ) = α k j ( α k r ( x ) ) = t q + ( 1 − t ) q k j α k r ( x ) − h 1 − q + h 1 − q < a − h 1 − q + h 1 − q = a ,

and hence, a − α k ( j + r ) ( x ) > 0 for all j ∈ N 0 . Note also that for any x , α k j ( x ) → h 1 − q as j → ∞ , and hence, a − α k j ( x ) ≤ 0 may hold only for finitely many j ∈ N . The rest proceeds similar to the proof of [10, Proposition 6].□

For m ∈ N 0 , the α k -factorial [1] is introduced as follows:

[ m ] α k ! ≔ [ m ] α k [ m − 1 ] α k [ m − 2 ] α k … [ 2 ] α k [ 1 ] α k

with convention [ 0 ] α k ! = 1 . For r ∈ R , m ∈ N , the α k -permutation coefficient [10] and the α k -binomial coefficient are defined, respectively, by

P α k [ r , m ] ≔ [ r ] α k [ r − 1 ] α k … [ r − m + 1 ] α k if m ∈ N , 1 if m = 0

and

r m α k ≔ P α k [ r , m ] [ m ] α k ! if m ∈ N , 1 if m = 0 .

We present the properties of the power function (11) whose proofs are also similar to the proofs of the properties of ( a + x ) α k r presented in [10].

Proposition 2.6

The following additivity rule holds for (11)
(12) ( a − x ) α k s + r = ( a − x ) α k s ( a − α k s ( x ) ) α k r .
The mth-order α k -derivative of (11) has the form
D α k m ( a − x ) α k r = ( − 1 ) m t q + ( 1 − t ) q k m ( m − 1 ) 2 P α k [ r , m ] ( a − α k m ( x ) ) α k r − m .
If m is a nonnegative integer, (11) produces the polynomial
( a − x ) α k m = ∏ j = 0 m − 1 ( a − α k j ( x ) ) .

3 α k -Power series and α k -Taylor series

Definition 3.1

Let c , c n ∈ R , then we define the α k -power series as follows:

∑ n = 0 ∞ c n ( x − c ) α k n .

Proposition 3.2

If t q + ( 1 − t ) q k < 1 , then to any α k -power series ∑ n = 0 ∞ c n ( x − c ) α k n , there corresponds a radius of convergence R , 0 ≤ R ≤ ∞ , such that the series is absolutely convergent for all x ∈ { x ∈ R : x − h 1 − q < R } ∪ { α k n ( c ) : n ∈ N 0 } and divergent for all x ∈ { x ∈ R : x − h 1 − q > R } \ { α k n ( c ) : n ∈ N 0 } . Moreover, for any 0 < r < R , the series is uniformly convergent on x − h 1 − q ≤ r . The radius R is calculated from the relations

1 R = limsup n → ∞ ∣ c n ∣ 1 n or 1 R = lim n → ∞ c n + 1 c n ,

whenever the latter exists.

Proof

Let a n = c n ( x − c ) α k n . Note that

∣ a n ∣ 1 n = ∣ c n ∣ 1 n ∣ ( x − c ) α k n ∣ 1 n .

Since t q + ( 1 − t ) q k < 1 , by (12) and Proposition 2.1 (vi), for x ∉ { α k n ( c ) } n ∈ N 0 , we have

lim n → ∞ ∣ ( x − c ) α k n + 1 ∣ ∣ ( x − c ) α k n ∣ = lim n → ∞ ∣ x − α k n ( c ) ∣ = x − h 1 − q ,

which implies lim n → ∞ ∣ ( x − c ) α k n ∣ 1 n = x − h 1 − q [14, 3.37 Theorem], and hence, [15, p. 194]

limsup n → ∞ ∣ a n ∣ 1 n = x − h 1 − q limsup n → ∞ ∣ c n ∣ 1 n = x − h 1 − q R .

Therefore, by the root test, the power series is convergent if x − h 1 − q < R and divergent if x − h 1 − q > R [16, Theorem 9].

Now, let 0 < r < R . We’ll prove that ∑ n = 0 ∞ c n ( x − c ) α k n converges uniformly on x − h 1 − q ≤ r . We’ll prove it by showing the tail, ∑ n = N ∞ c n ( x − c ) α k n , of the power series can be made arbitrarily small for large enough N ’s that can be chosen independent of x . First, we choose ρ < R and δ > 0 satisfying r + δ ≤ ρ and then we choose n 0 such that ∣ α k n ( c ) − h 1 − q ∣ < δ for all n ≥ n 0 . So, if x − h 1 − q ≤ r and n ≥ n 0 , then

(13) ∣ x − α k n ( c ) ∣ ≤ x − h 1 − q + α k n ( c ) − h 1 − q < r + δ ≤ ρ .

For any N ≥ n 0 , we have

∑ n = N ∞ c n ( x − c ) α k n = ∣ ( x − c ) α k n 0 ∣ ∑ n = N ∞ c n ( x − α k n 0 ( c ) ) α n − n 0 .

Since ∣ ( x − c ) α k n 0 ∣ is continuous, it is uniformly bounded on the compact set x − h 1 − q ≤ r , say bounded by M , and so by (13), we have

(14) ∑ n = N ∞ c n ( x − c ) α k n ≤ M ρ n 0 ∑ n = N ∞ ∣ c n ∣ ρ n .

Since ρ < R , the usual power series ∑ n = 0 ∞ c n ρ n converges absolutely and the right-hand side of (14) can be made arbitrarily small by choosing N large enough, and we are done.□

Proposition 3.3

For t q + ( 1 − t ) q k < 1 , let f ( x ) = ∑ n = 0 ∞ c n ( x − c ) α k n and R > 0 be its radius of convergence. Then for 0 < x − h 1 − q < R ,

(15) D α k f ( x ) = ∑ n = 1 ∞ c n [ n ] α k ( x − c ) α k n − 1 .

Proof

Suppose 0 < x − h 1 − q < R . First note that Proposition 2.1 (iv) implies

α k ( x ) − h 1 − q = t q + ( 1 − t ) q k x − h 1 − q < x − h 1 − q < R .

So, f ( α k ( x ) ) = ∑ n = 0 ∞ c n ( α k ( x ) − c ) α k n converges absolutely, and hence, by power rule for derivatives (10), we have

□ D α k f ( x ) = ∑ n = 0 ∞ c n ( α k ( x ) − c ) α k n − ∑ n = 0 ∞ c n ( x − c ) α k n α k ( x ) − x = ∑ n = 1 ∞ c n ( α k ( x ) − c ) α k n − ( x − c ) α k n α k ( x ) − x = ∑ n = 1 ∞ c n D α k ( x − c ) α k n = ∑ n = 1 ∞ c n [ n ] α k ( x − c ) α k n − 1 .

Remark 3.4

Note that, when t q + ( 1 − t ) q k < 1 , we have lim n → ∞ [ n + 1 ] α k [ n ] α k = 1 , and so

limsup n → ∞ ∣ c n ∣ 1 n [ n ] α k 1 n = limsup n → ∞ ∣ c n ∣ 1 n lim n → ∞ [ n ] α k 1 n = limsup n → ∞ ∣ c n ∣ 1 n lim n → ∞ [ n + 1 ] α k [ n ] α k = limsup n → ∞ ∣ c n ∣ 1 n .

Therefore, ∑ n = 1 ∞ c n [ n ] α k ( x − c ) α k n − 1 and ∑ n = 1 ∞ c n ( x − c ) α k n have the same radius of convergence.

The following proposition shows that (15) also holds for x = h 1 − q .

Proposition 3.5

For t q + ( 1 − t ) q k < 1 , let f ( x ) = ∑ n = 0 ∞ c n ( x − c ) α k n and R > 0 be its radius of convergence. Then for any 0 < r < R

∑ n = 1 ∞ d d x ( c n ( x − c ) α k n )

converges uniformly on x − h 1 − q ≤ r , and hence,

d d x ∑ n = 0 ∞ c n ( x − c ) α k n = ∑ n = 1 ∞ d d x ( c n ( x − c ) α k n ) ,

in particular,

( D α k f ) h 1 − q = ∑ n = 1 ∞ c n [ n ] α k h 1 − q − c α k n − 1 .

Proof

Let 0 < r < R and choose ρ such that r < ρ < R . Since limsup n → ∞ ∣ n c n ∣ 1 n = limsup n → ∞ ∣ c n ∣ 1 n , the series ∑ n = 0 ∞ n c n ρ n is absolutely convergent, and so, it is enough to show that there exists M > 0 , N > 0 such that

d d x ( x − c ) α k n ≤ M n ρ n for all x − h 1 − q ≤ r and for all n ≥ N .

By applying the product rule for the usual derivative, we obtain

d d x ( x − c ) α k n = ∑ j = 0 n − 1 u j , n ( x ) where u j , n ( x ) = ∏ i = 0 i ≠ j n − 1 ( x − α k i ( c ) ) .

By Proposition 2.1 (vi), we choose an N ∈ N so that ∣ h 1 − q − α k i ( c ) ∣ < ( ρ − r ) for all i ≥ N , and set

M = max M 0 ρ N , M 1 ρ N , M 2 ρ N , … , M N − 1 ρ N , M N ρ N + 1 ,

where

M j = max x − h 1 − q ≤ r ∏ i = 0 i ≠ j N − 1 ∣ x − α k i ( c ) ∣ for 0 ≤ j ≤ N − 1 and M N = max x − h 1 − q ≤ r ∏ i = 0 N − 1 ∣ x − α k i ( c ) ∣ .

Now suppose n ≥ N and x − h 1 − q < r . If i ≥ N , then

∣ x − α k i ( c ) ∣ ≤ x − h 1 − q + h 1 − q − α k i ( c ) ≤ r + ( ρ − r ) = ρ ,

and hence, for j < N , we have

∣ u j , n ( x ) ∣ = ∏ i = 0 i ≠ j N − 1 ∣ x − α k i ( c ) ∣ ∏ i = N n − 1 ∣ x − α k i ( c ) ∣ ≤ M j ρ n − N ≤ M ρ n ,

and for j ≥ N , we have

∣ u j , n ( x ) ∣ = ∏ i = 0 N − 1 ∣ x − α k i ( c ) ∣ ∏ i = N i ≠ j n − 1 ∣ x − α k i ( c ) ∣ ≤ M N ρ n − N − 1 ≤ M ρ n .

Therefore,

d d x ( x − c ) α k n ≤ ∑ j = 0 n − 1 ∣ u j , n ( x ) ∣ ≤ M n ρ n ,

and we are done.□

As a result of Propositions 3.3 and 3.5, we conclude with the uniqueness of power series representations.

Corollary 3.6

If f ( x ) = ∑ n = 0 ∞ c n ( x − c ) α k n on x − h 1 − q < R for some R > 0 , then

c n = ( D α n f ) ( c ) [ n ] α ! for a l l n ∈ N 0 .

Remark 3.7

To make convergence issues easier, it is convenient to deal with the series

∑ n = 0 ∞ c n x − h 1 − q n .

Definition 3.8

We define the α k -Taylor series of a function f at x = c by the series

∑ j = 0 ∞ ( D α k j f ) ( c ) [ j ] α k ! ( x − c ) α k j .

If c = h 1 − q , then the series is said to be the α k -Maclaurin series of f . If f equals its α k -Taylor series in an interval, then f is said to be α k -analytic on that interval.

Example 3.9

Since P α k [ n , j ] = 0 for all j > n , the Taylor series of ( x − y ) α k n at a is the finite sum [1, Theorem 4]

(16) ( x − y ) α k n = ∑ j = 0 n n j α k ( a − y ) α k n − j ( x − a ) α k j .

The sum (16) is the α -analogue of Gauss’s binomial formula inspiring the additivity of exponential function, which will be demonstrated in the forthcoming section.

Example 3.10

We obtain the Maclaurin series of ( a − x ) α k r as follows:

∑ j = 0 ∞ ( − 1 ) j t q + ( 1 − t ) q k j ( j − 1 ) 2 r j α k a − h 1 − q r − j x − h 1 − q j .

It is possible to show that both the aforementioned Maclaurin series and ( a − x ) α k r satify the IVP

( a − x ) D α k y ( x ) = − [ r ] α k y ( x ) , y h 1 − q = a − h 1 − q r ,

and the series is absolutely convergent for x satisfying x − h 1 − q < ∣ a − h 1 − q ∣ t q + ( 1 − t ) q k r . Hence, by the uniqueness of the solution of the IVP, we end up with

(17) ( a − x ) α k r = ∑ j = 0 ∞ ( − 1 ) j t q + ( 1 − t ) q k j ( j − 1 ) 2 r j α k a − h 1 − q r − j x − h 1 − q j .

In particular, if a = 1 + h 1 − q , for h 1 − q ≤ x ≤ 1 + h 1 − q , we obtain

1 + h 1 − q − x α k r = ∑ j = 0 ∞ ( − 1 ) j t q + ( 1 − t ) q k j ( j − 1 ) 2 r j α k x − h 1 − q j .

Remark 3.11

It is difficult to obtain the derivative of ( x − a ) α k r from the definition, and there is no obvious relation with ( a − x ) α k r like Proposition 2.3 when r is not an integer. Nevertheless, by using the previous example, we can show that its derivative also obeys the power rule for derivative.

Proposition 3.12

For r ∈ R , we have

(18) D α k ( x − a ) α k r = [ r ] α k ( x − a ) α k r − 1 .

Proof

Indeed, by (17) and Proposition 2.3, we obtain

( x − a ) α k r = ∑ j = 0 ∞ ( − 1 ) j t q + ( 1 − t ) q k j ( j − 1 ) 2 r j α k x − h 1 − q r − j a − h 1 − q j , = ∑ j = 0 ∞ r j α k x − h 1 − q r − j h 1 − q − a α k j .

Noting that D α k ( x − h 1 − q ) s = [ s ] α k ( x − h 1 − q ) s − 1 for all s ∈ R \ { 0 } and then differentiating termwise results

□ D α k ( x − a ) α k r = ∑ j = 0 ∞ r j α k [ r − j ] α k x − h 1 − q r − j − 1 h 1 − q − a α k j = ∑ j = 0 ∞ [ r ] α k r − 1 j α k x − h 1 − q ( r − 1 ) − j h 1 − q − a α k j = [ r ] α k ( x − a ) α k r − 1 .

4 α k -Exponential function and α k -gamma function

Definition 4.1

Let c be any nonzero constant and t q + ( 1 − t ) q k < 1 . Then we introduce the α k -exponential function by

(19) Exp α k ( c ( x − a ) ) ≔ ∑ j = 0 ∞ c j ( x − a ) α k j [ j ] α k ! .

We denote Exp α k ( 1 ( x − a ) ) by Exp α k ( x − a ) .

For a = h 1 − q , the α k -exponential function (19) turns out to be

Exp α k x − h 1 − q = ∑ j = 0 ∞ x − h 1 − q j [ j ] α k ! .

Proposition 4.2

The α k -exponential function (19) converges absolutely when x − h 1 − q < 1 ∣ c ∣ 1 − t q + ( 1 − t ) q k or x = α k n ( a ) for some n ∈ N 0 , and diverges when x − h 1 − q > 1 ∣ c ∣ 1 − t q + ( 1 − t ) q k and x ≠ α k n ( a ) for some n ∈ N 0 . Moreover, for any 0 < r < 1 ∣ c ∣ 1 − t q + ( 1 − t ) q k , the series is uniformly convergent on x − h 1 − q ≤ r . In particular, Exp α k ( x − h 1 − q ) converges absolutely on x − h 1 − q < 1 1 − t q + ( 1 − t ) q k and diverges on x − h 1 − q ≥ 1 1 − t q + ( 1 − t ) q k .

Proof

Let c j = c j [ j ] α k ! . Then by Proposition 2.1 (iii),

lim j → ∞ ∣ c j + 1 ∣ ∣ c j ∣ = lim j → ∞ ∣ c ∣ [ j + 1 ] α k = ∣ c ∣ 1 − t q + ( 1 − t ) q k .

If we show that Exp α k ( x − h 1 − q ) diverges when x − h 1 − q = 1 1 − t q + ( 1 − t ) q k , the rest follows from Proposition 3.2. If x − h 1 − q = 1 1 − t q + ( 1 − t ) q k and c = 1 , then

∣ c j + 1 ( x − h 1 − q ) j + 1 ∣ ∣ c j ( x − h 1 − q ) j ∣ = 1 1 − t q + ( 1 − t ) q k ( j + 1 ) > 1 ,

and so ∣ c j ( x − h 1 − q ) j ∣ > ∣ c 0 ( x − h 1 − q ) 0 ∣ = 1 for all j ∈ N 0 . This shows that lim j → ∞ c j ( x − h 1 − q ) j ≠ 0 , and hence, Exp α k ( x − h 1 − q ) diverges by the general term test.□

Proposition 4.3

The α k -exponential function (19) is the unique solution of the IVP

D α k y ( x ) = c y ( x ) , y ( a ) = 1 .

Proof

It is obvious from the definition that Exp α k ( c ( a − a ) ) = 1 . To prove

D α k Exp α k ( c ( x − a ) ) = c Exp α k ( c ( x − a ) ) ,

we use Proposition 3.3 as follows:

□ D α k Exp α k ( c ( x − a ) ) = ∑ n = 1 ∞ c n [ n ] α k ( x − a ) α k n − 1 [ n ] α k ! = ∑ n = 1 ∞ c n ( x − a ) α k n − 1 [ n − 1 ] α k ! = c ∑ n = 0 ∞ c n ( x − a ) α k n [ n ] α k ! = c Exp α k ( c ( x − a ) ) .

To visualize the α -exponential function, we refer to Figure 1, which can be considered as approximations to discrete exponential functions.

$Figure 1 The α k {\alpha }^{k} -exponential function.$

Figure 1

The α k -exponential function.

Proposition 4.4

The α -analogue of additive identity of exponential functions takes the form

Exp α k ( x − y ) = Exp α k ( x − a ) Exp α k ( a − y ) .

In particular,

(20) Exp α k ( x − a ) Exp α k ( a − x ) = 1 .

Proof

Note that

Exp α k ( x − a ) Exp α k ( a − y ) = ∑ j = 0 ∞ ( x − a ) α k j [ j ] α k ! ∑ ℓ = 0 ∞ ( a − y ) α k ℓ [ ℓ ] α k ! = ∑ j = 0 ∞ ∑ ℓ = 0 ∞ j + ℓ j α k ( x − a ) α k j ( a − y ) α k ℓ [ j + ℓ ] α k ! .

We reindex the double sum by putting j + ℓ = n and use Example 3.9 to obtain

□ Exp α k ( x − a ) Exp α k ( a − y ) = ∑ n = 0 ∞ 1 [ n ] α k ! ∑ j = 0 n n j α k ( x − a ) α k j ( a − y ) α k n − j = ∑ n = 0 ∞ ( x − y ) α k n [ n ] α k ! = Exp α k ( x − y ) .

Proposition 4.5

Exp α k ( a − x ) has the following series representation

Exp α k ( a − x ) = ∑ j = 0 ∞ ( − 1 ) j t q + ( 1 − t ) q k ( j − 1 ) j 2 ( x − a ) α − k j [ j ] α k ! = ∑ j = 0 ∞ ( − 1 ) j ( x − a ) α − k j [ j ] α − k ! .

In particular,

Exp α k h 1 − q − x = ∑ j = 0 ∞ ( − 1 ) j t q + ( 1 − t ) q k ( j − 1 ) j 2 x − h 1 − q j [ j ] α k ! .

The proof is a direct consequence of Proposition 2.3.

Remark 4.6

One of the main advantages of studying on α -time scale is that it allows unification and extension of nabla and delta ( q , h ) -time scales. We will observe that for t ∈ { 0 , 1 } , α -exponential function (19) recovers the nabla ( q , h ) -exponential function [12] and the delta ( q , h ) -exponential function [17] as well as q -exponential [13] and h -exponential functions [3], while for t ∈ ( 0 , 1 ) , it produces their extensions. The related analysis can be figured out as follows, where for simplicity, we fix k = 1 .

As ( q , h ) → ( 1 , 0 ) and a = 0 , the α -exponential function (19) recovers the ordinary exponential function.
If t = 0 , by definition of α -operator (1), α j ( x ) = q j x + h [ j ] q and if 0 < q < 1 , x 0 < h 1 − q or q > 1 , x 0 > h 1 − q , we obtain α j ( x ) = σ j ( x ) , where σ is the forward jump operator on ( q , h ) -time scale. Then the α -polynomial (9) and the α -exponential function (19) produce the delta ( q , h ) -polynomial
(21) ( x − a ) q , h m = ∏ j = 0 m − 1 ( x − ( q j a + h [ j ] q ) )
and the delta ( q , h ) -exponential function
(22) Exp ( q , h ) ( x − a ) = ∑ j = 0 ∞ ( x − a ) q , h j [ j ] q ! ,
respectively, which are studied in [17].
1. If also h = 0 and a = 0 , then (21) and (22), respectively, imply ( x − 0 ) q , 0 m = x m and the q -exponential function [13]
  e q x = ∑ j = 0 ∞ x j [ j ] q ! .
2. If also q = 1 and a = 0 , then (21) implies that
  ( x − 0 ) 1 , h m = x ( x − h ) … ( x − ( m − 1 ) h ) = x h x h − 1 … x h − ( m − 1 ) h m
  which allows the h -exponential function
  ∑ j = 0 ∞ ( x − 0 ) 1 , h j j ! = ∑ j = 0 ∞ x h j h j = ( 1 + h ) x h .
  For details of delta ( q , h ) -, q -, h -exponential functions, we refer to [17, Remark 4.8].
If t = 1 , by definition of α -operator (1), α j ( x ) = x − h [ j ] q q j and if q > 1 , x 0 > h 1 − q , we obtain α j ( x ) = ρ j ( x ) , where ρ is the backward jump operator on ( q , h ) -time scale. Then the α -polynomial (9) and the α -exponential function (19) reduce to the nabla ( q , h ) -polynomial
(23) ( x − a ) q , h ˜ m = ∏ j = 0 m − 1 x − a − h [ j ] q q j ,
and the nabla ( q , h ) -exponential function
(24) Exp ˜ ( q , h ) ( x − a ) = ∑ j = 0 ∞ ( x − a ) q , h ˜ j [ j ] 1 q ! ,
respectively, which are presented in [12].
1. If also h = 0 and a = 0 , then (23) and (24) lead to, respectively, ( x − 0 ) q , 0 ˜ m = x m and the Euler’s q -exponential function [13]
  ∑ j = 0 ∞ x j [ j ] 1 q ! = ∑ j = 0 ∞ q j ( j − 1 ) 2 x j [ j ] q ! = E q x .
2. If also q = 1 and a = 0 , then (23) implies that
  ( x − 0 ) 1 , h ˜ m = x ( x + h ) … ( x + ( m − 1 ) h ) = − x h − x h − 1 … − x h − ( m − 1 ) ( − h ) m ,
  which allows the h -exponential function
  ∑ j = 0 ∞ − x h j ( − h ) j = ( 1 − h ) − x h .
For details of nabla ( q , h ) -, q -, h -exponential functions, we refer to [12, Remark 4.5].
Consider t = q 2 − q s + 1 q 2 − 1 for ∣ s ∣ < 1 , then t ∈ ( 0 , 1 ) and α m ( x ) = q s m x + h [ s ] q [ m ] α .
(25) ∏ j = 0 m − 1 ( x − q s j a − h [ s ] q [ j ] α ) , ∣ s ∣ < 1 .
When 0 < s < 1 , the polynomial (25) reclaims the extensions of delta ( q , h ) -polynomial (21) and delta ( q , h ) -exponential function (22), while for − 1 < s < 0 , it produces the extensions of nabla ( q , h ) -polynomial (23) and nabla ( q , h ) -exponential function (24).

Definition 4.7

For t q + ( 1 − t ) q k < 1 and s > 0 , we define the α k -gamma function as follows:

(26) Γ α k ( s ) ≔ ∫ ( h 1 − q , [ ∞ ] α k ] x − h 1 − q s − 1 Exp α k h 1 − q − α k ( x ) d α k x ,

where [ ∞ ] α k = h 1 − q + 1 1 − t q + ( 1 − t ) q k .

Remark 4.8

The reason why we choose the upper limit of the integral as h 1 − q + 1 1 − t q + ( 1 − t ) q k is that the relation (20) holds when both series Exp α k ( x − a ) and Exp α k ( a − x ) converge, and hence, when x − h 1 − q < 1 1 − t q + ( 1 − t ) q k . Note also that by Proposition 4.2, Exp α k [ ∞ ] α k − h 1 − q = ∞ as a divergent series with nonnegative terms and hence Exp α k h 1 − q − [ ∞ ] α k = 0 by (20).

Remark 4.9

As the α -polynomial and α -exponential function, the α -gamma function (26) has the following “nice” reductions.

As ( q , h ) → ( 1 , 0 ) , the α -gamma function (26) recovers the ordinary gamma function.
Up to our knowledge, delta and nabla ( q , h ) -gamma functions have not been studied so far. In the light of α -gamma function (26), we are able to provide delta and nabla ( q , h ) -gamma functions.
1. Indeed, the α -gamma function (26) provides the delta ( q , h ) -gamma function
  (27) ∫ h 1 − q h + 1 1 − q x − h 1 − q s − 1 Exp ( q , h ) h 1 − q − σ ( x ) d ( q , h ) x ,
  where Exp ( q , h ) h 1 − q − σ ( x ) = ∑ j = 0 ∞ h 1 − q − σ ( x ) q , h j [ j ] q ! . Furthermore, (27) produces the delta q -gamma function [18]
  Γ q ( s ) = ∫ 0 1 1 − q x s − 1 E q − q x d q x ,
  since Exp ( q , 0 ) ( 0 − σ ( x ) ) = ∑ j = 0 ∞ ( 0 − q x ) ( q , 0 ) j [ j ] q ! = ∑ j = 0 ∞ ( − 1 ) j q j ( j + 1 ) 2 x j [ j ] q ! = ∑ j = 0 ∞ q j ( j − 1 ) 2 ( − q x ) j [ j ] q ! = E q − q x .
2. The nabla ( q , h ) -power function and its properties are presented in [19], and now, the α -gamma function leads to the nabla ( q , h ) -gamma function
  (28) ∫ h 1 − q h − q 1 − q x − h 1 − q s − 1 Exp ˜ ( q , h ) h 1 − q − ρ ( x ) d ˜ ( q , h ) x ,
  where Exp ˜ ( q , h ) h 1 − q − ρ ( x ) = ∑ j = 0 ∞ h 1 − q − ρ ( x ) q , h ˜ j [ j ] 1 q ! .
By a similar fashion, equipped with (25), it is possible to extend the delta ( q , h ) -gamma function (27) and nabla ( q , h ) -gamma function (28).

Theorem 4.10

The α k -gamma function satisfies

Γ α k ( 1 ) = 1 ,
Γ α k ( s + 1 ) = [ s ] α k Γ α k ( s ) , and hence, Γ α k ( n + 1 ) = [ n ] α k ! for n ∈ N 0 .

Proof

(i) Note that the quotient rule (6), Propositions 4.3 and 4.4 imply that

D α k Exp α k ( a − x ) = D α k 1 Exp α k ( x − a ) = − Exp α k ( x − a ) Exp α k ( x − a ) Exp α k ( α k ( x ) − a ) = − Exp α k ( a − α k ( x ) ) ,

and hence by (7), we obtain

Γ α k ( 1 ) = ∫ ( h 1 − q , [ ∞ ] α k ] Exp α k h 1 − q − α k ( x ) d α k x = − Exp α k h 1 − q − x ∣ h 1 − q [ ∞ ] α k = 0 + 1 = 1 .

(ii) Applying Theorem 2.2, we obtain

□ Γ α k ( s + 1 ) = ∫ ( h 1 − q , [ ∞ ] α k ] x − h 1 − q s Exp α k h 1 − q − α k ( x ) d α k x = − x − h 1 − q s Exp α k h 1 − q − x h 1 − q [ ∞ ] α k + [ s ] α k ∫ ( h 1 − q , [ ∞ ] α k ] x − h 1 − q s − 1 Exp α k h 1 − q − α k ( x ) d α k x = [ s ] α k Γ α k ( s ) .

For − 1 < s < 0 , we define

Γ α k ( s ) = Γ α k ( s + 1 ) [ s ] α k ,

so Theorem 4.10 (ii) remains valid. Inductively, for n ∈ N and − n − 1 < s < − n , we define Γ α k ( s ) in a similar fashion.

Remark 4.11

By Theorem 4.10, we can deduce that α k -gamma function solves the interpolation problem [20]

(29) f ( s + 1 ) = ϕ ( s ) f ( s ) , f ( 1 ) = 1 ,

with ϕ ( s ) = [ s ] α k . In [21], the gamma function on time scales was studied by using Laplace transform where the functional relation (29) holds for particular time scales such as the set of reals and quantum numbers.

The Bohr-Mollerup theorem [14] is crucial in mathematical analysis as it characterizes the gamma function as the unique positive, logarithmically convex function on ( 0 , ∞ ) satisfying (29) with ϕ ( s ) = s . Now we prove that the α k -gamma function satisfies the α -version of Bohr-Mollerup theorem. To present this significant result, we first state and prove the logarithmic convexity of α k -gamma function.

Lemma 4.12

If p , q > 0 with 1 p + 1 q = 1 , then the α k -gamma function (26) admits the following inequality:

(30) Γ α k x p + y q ≤ ( Γ α k ( x ) ) 1 p ( Γ α k ( y ) ) 1 q .

Proof

Note that when 1 p + 1 q = 1 , we have

τ − h 1 − q x p + y q − 1 = τ − h 1 − q x − 1 p τ − h 1 − q y − 1 q

and

Exp α k h 1 − q − α k ( τ ) = Exp α k h 1 − q − α k ( τ ) 1 ⁄ p Exp α k h 1 − q − α k ( τ ) 1 ⁄ q .

Hence, by Hölder’s inequality, we deduce that

□ Γ α k x p + y q = ∫ ( h 1 − q , [ ∞ ] α k ] τ − h 1 − q x p + y q − 1 Exp α k h 1 − q − α k ( τ ) d α k τ ≤ ∫ ( h 1 − q , [ ∞ ] α k ] τ − h 1 − q x − 1 Exp α k h 1 − q − α k ( τ ) d α k τ 1 ⁄ p × ∫ ( h 1 − q , [ ∞ ] α k ] τ − h 1 − q y − 1 Exp α k h 1 − q − α k ( τ ) d α k τ 1 ⁄ q = ( Γ α k ( x ) ) 1 ⁄ p ( Γ α k ( y ) ) 1 ⁄ q .

Note that the logarithmic convexity of the α k -gamma function (26) follows directly from taking the natural logarithm of (30).

Theorem 4.13

If f is a positive function on ( 0 , ∞ ) satisfying the conditions

f ( s + 1 ) = [ s ] α k f ( s ) ,
f ( 1 ) = 1 ,
log f is convex,

then f ( s ) = Γ α k ( s ) .

Proof

By (i), it is enough to prove the theorem for s ∈ ( 0 , 1 ) . Let ψ = log f . Then ψ is convex, and hence, for any x 1 , x 2 > 0 and 0 < λ < 1 , we have

(31) ψ ( λ x 1 + ( 1 − λ ) x 2 ) ≤ λ ψ ( x 1 ) + ( 1 − λ ) ψ ( x 2 ) .

By putting x 1 = n , x 2 = n + 1 + s and λ = s s + 1 in (31), we obtain

ψ ( n + 1 ) ≤ s s + 1 ψ ( n ) + 1 s + 1 ψ ( n + 1 + s ) ,

from which we deduce

(32) log [ n ] α k ≤ ψ ( n + 1 + s ) − ψ ( n + 1 ) s .

By putting x 1 = n + 1 , x 2 = n + 2 and λ = 1 − s in (31), we obtain

ψ ( n + 1 + s ) ≤ ( 1 − s ) ψ ( n + 1 ) + s ψ ( n + 2 ) ,

which yields to the inequality

(33) ψ ( n + 1 + s ) − ψ ( n + 1 ) s ≤ log [ n + 1 ] α k .

By repeated use of (i), we obtain

(34) ψ ( n + 1 ) = log ( [ n ] α k ! ) and ψ ( n + 1 + s ) = ψ ( s ) + log ( [ s ] α k … [ s + n ] α k ) .

Together with (32) and (33), (34) implies

log [ n ] α k ≤ ψ ( s ) + log ( [ s ] α k … [ s + n ] α k ) − log ( [ n ] α k ! ) s ≤ log [ n + 1 ] α k ,

and hence,

0 ≤ ψ ( s ) − log [ n ] α k ! [ n ] α k s [ s ] α k … [ s + n ] α k ≤ s log [ n + 1 ] α k [ n ] α k .

The last expression tends to 0 as n → ∞ . Therefore, ψ ( s ) is uniquely determined, which completes the proof.□

Remark 4.14

The last line of the proof provides another form for α k -gamma function

(35) Γ α k ( s ) = lim n → ∞ [ n ] α k ! [ n ] α k s [ s ] α k [ s + 1 ] α k … [ s + n ] α k for s ∈ ( 0 , 1 ) .

Since Γ α k ( s + 1 ) = [ s ] α k Γ α k ( s ) , one can deduce that the product form (35) of the α k -gamma function (26) holds indeed for all s > 0 .

Proposition 4.15

The α k -gamma function (26) satisfies the following asymptotic properties:

lim s → 0 + Γ α k ( s ) = ∞ and lim s → ∞ Γ α k ( s ) = ∞ .

Proof

As a logarithmically convex function, the α k -gamma function (26) is continuous on ( 0 , ∞ ) [14, p. 101] and hence lim s → 0 Γ α k ( s + 1 ) = Γ α k ( 1 ) = [ 1 ] α k = 1 . Since lim s → 0 [ s ] α k = 0 and [ s ] α k > 0 for s > 0 , the relation Γ α k ( s + 1 ) = [ s ] α k Γ α k ( s ) implies lim s → 0 + Γ α k ( s ) = ∞ . For the second limit, let s > 3 . By using the logarithmic convexity of the α k -gamma function, we have

log Γ α k ( 3 ) = log Γ α k s − 3 s − 1 ⋅ 1 + 3 − 1 s − 1 ⋅ s ≤ s − 3 s − 1 log Γ α k ( 1 ) + 3 − 1 s − 1 log Γ α k ( s ) = 2 s − 1 log Γ α k ( s ) .

Since Γ α k ( 3 ) = 1 + t q + ( 1 − t ) q > 1 , we have log Γ α k ( 3 ) > 0 , and so lim s → ∞ ( s − 1 ) log Γ α k ( 3 ) = ∞ . Therefore, lim s → ∞ log Γ α k ( s ) = ∞ , and hence, lim s → ∞ Γ α k ( s ) = ∞ .□

To visualize the α k -gamma function, we refer to Figure 2.

$Figure 2 The graph of α k {\alpha }^{k} -gamma function.$

Figure 2

The graph of α k -gamma function.

5 α -Cylindrical diffusion equation

In this section, for simplicity, we fix k = 1 . In the literature, Bessel functions appear across numerous disciplines, including applied mathematics, physics, engineering, biology, probability, geophysics, and even signal processing. They are vital for facilitating the analytical representation of wave functions and potential distributions within cylindrical and spherical geometries. A significant application of Bessel functions lies in the analytical solution of the diffusion equation in cylindrical coordinates.

Our primary goal is to introduce an α -partial difference equation whose solutions encompass the α -exponential and α -gamma functions, as well as provide α -analogues of Bessel and modified Bessel functions. These functions will be constructed as solutions of the α -analogues of the Bessel and modified Bessel equations respectively embedded within the following α -partial difference equation. To facilitate this, we propose the α -analogue of the cylindrical diffusion equation as follows:

(36) t q + ( 1 − t ) q ( x − a ) α 2 ∂ α , x 2 z ( x , τ ; α 2 ( a ) ) + ( x − a ) α 1 ∂ α , x z ( x , τ ; α ( a ) ) = 1 K ( x − a ) α 2 ∂ α , τ z ( x , τ ; α 2 ( a ) ) ,

where z ( x , τ ; a ) is the temperature of the cylinder at time τ and radius x and K is the thermal diffusivity constant. It is clear that when a = 0 and as ( q , h ) → ( 1 , 0 ) , (36) recovers the cylindrical diffusion equation [22]

∂ z ∂ τ = K ∂ 2 z ∂ x 2 + 1 x ∂ z ∂ x .

To solve (36), we utilize the separation of variables method to write z ( x , τ ; α j ( a ) ) = u ( x ; α j ( a ) ) w ( τ ; a ) for j = 1 , 2 . Then, we have the α -partial derivatives of z as follows:

∂ α , x j z ( x , τ ; α j ( a ) ) = D α j u ( x ; α j ( a ) ) w ( τ ; a ) , j = 1 , 2 , ∂ α , τ z ( x , τ ; α 2 ( a ) ) = u ( x ; α 2 ( a ) ) D α w ( τ ; a ) ,

which imply

t q + ( 1 − t ) q ( x − a ) α 2 D α 2 u ( x ; α 2 ( a ) ) w ( τ ; a ) + ( x − a ) α 1 D α u ( x ; α ( a ) ) w ( τ ; a ) = 1 K ( x − a ) α 2 D α w ( τ ; a ) u ( x ; α 2 ( a ) ) ,

from which we derive the following equation by dividing with ( x − a ) α 2 u ( x ; α 2 ( a ) ) w ( τ ; a ) ,

t q + ( 1 − t ) q ( x − a ) α 2 D α 2 u ( x ; α 2 ( a ) ) + ( x − a ) α 1 D α u ( x ; α ( a ) ) ( x − a ) α 2 u ( x ; α 2 ( a ) ) = 1 K D α w ( τ ; a ) w ( τ ; a ) = − η .

As a consequence, η is constant, and we obtain two α -ordinary difference equations:

(37) D α w ( τ ; a ) + η K w ( τ ; a ) = 0 ,

(38) t q + ( 1 − t ) q ( x − a ) α 2 D α 2 u ( x ; α 2 ( a ) ) + ( x − a ) α 1 D α u ( x ; α ( a ) ) + η ( x − a ) α 2 u ( x ; α 2 ( a ) ) = 0 .

We analyze equations (37) and (38) for different values of η .

Case I. η = 0 : If w ( τ ; a ) = c 1 , where c 1 is a constant, then D α w ( τ ; a ) = 0 . On the other hand, (38) yields to

t q + ( 1 − t ) q ( x − a ) α 2 D α 2 u ( x ; α 2 ( a ) ) + ( x − a ) α 1 D α u ( x ; α ( a ) ) = 0 ,

which turns out to be

(39) ( x − a ) t q + ( 1 − t ) q ( x − α ( a ) ) D α 2 u ( x ; α 2 ( a ) ) + D α u ( x ; α ( a ) ) = ( x − a ) ( ( α ( x ) − α 2 ( a ) ) D α 2 u ( x ; α 2 ( a ) ) + D α u ( x ; α ( a ) ) ) = 0 ,

where we used the additivity rule (12), which implies ( x − a ) α 2 = ( x − a ) ( x − α ( a ) ) and Proposition 2.1 (viii) which implies t q + ( 1 − t ) q ( x − α ( a ) ) = α ( x ) − α 2 ( a ) . Assuming

(40) D α 2 u ( x ; α 2 ( a ) ) = D α 2 u ( x ; α ( a ) )

and using the product rule (5), (39) yields as follows:

D α ( ( x − α 2 ( a ) ) D α u ( x ; α ( a ) ) ) = 0 ,

which provides

(41) D α u ( x ; α ( a ) ) = C x − α 2 ( a ) ,

where C is a constant. Now, we need to offer an α -logarithm function, similar to the one in [10] as follows:

Log α ( x − a ) ≔ t q + ( 1 − t ) q − 1 ln t q + ( 1 − t ) q ln x − h 1 − q if a = h 1 − q , ∫ h 1 − q , x d α τ τ − a if a < h 1 − q .

Hence (41) yields as

u ( x ; a ) = C Log α ( x − a ) .

Note that, by Proposition 2.1 (v), the assumption (40) implies that a = h 1 − q . Therefore, we conclude that

u x ; h 1 − q = C t q + ( 1 − t ) q − 1 ln t q + ( 1 − t ) q ln x − h 1 − q .

Case II. η > 0 : Set η = λ 2 . By Proposition 4.3, we assure that w ( τ ; a ) = Exp α ( − λ 2 K ( τ − a ) ) solves the equation D α w ( τ ; a ) + λ 2 K w ( τ ; a ) = 0 . On the other hand, (38) implies the equation

(42) L ( u ) ≔ P ( x ; a ) D α 2 u ( x ; α 2 ( a ) ) + Q ( x ; a ) D α u ( x ; α ( a ) ) + R ( x ; a ) u ( x ; α 2 ( a ) ) = 0 ,

where

P ( x ; a ) = t q + ( 1 − t ) q ( x − a ) α 2 , Q ( x ; a ) = ( x − a ) α 1 , R ( x ; a ) = λ 2 ( x − a ) α 2 .

Notice that (42) covers the ordinary Bessel equation of order zero

x 2 u ″ + x u ′ + x 2 u = 0 ,

when λ = 1 , a = 0 and ( q , h ) → ( 1 , 0 ) . Therefore, when λ = 1 , (42) can be referred as α -analogue of Bessel equation of order zero.

Definition 5.1

We introduce the α -Bessel function of order zero of the first kind as follows:

(43) J 0 α ( x − a ) ≔ ∑ j = 0 ∞ ( − 1 ) j ( x − a ) α 2 j [ 2 ] α 2 j ( Γ α 2 ( j + 1 ) ) 2 ,

where Γ α 2 is the α 2 -gamma function introduced in Definition 4.7.

Notice that if a = 0 and as ( q , h ) → ( 1 , 0 ) , (43) recovers the ordinary Bessel function of order zero of the first kind. Moreover, h Z analogues of (42) and (43) are presented in [23], Z analogues of (42) and (43) are presented in [24], while q -analogue of (43) is extensively studied in the literature [25,26].

Theorem 5.2

A solution of the equation (42) is presented as follows:

(44) J 0 α ( λ ( x − a ) ) = ∑ j = 0 ∞ ( − 1 ) j λ 2 j ( x − a ) α 2 j [ 2 ] α 2 j ( Γ α 2 ( j + 1 ) ) 2

and converges absolutely when ∣ x − h 1 − q ∣ < 1 ∣ λ ∣ 1 − t q + ( 1 − t ) q .

Before the proof, we require the definition below.

Definition 5.3

We define x * as a singular point of (42) if P ( x * ; a ) = 0 . In addition, if the limits

lim x → x * ( x − x * ) α 1 Q ( x ; a ) P ( x ; a ) and lim x → x * ( x − x * ) α 2 R ( x ; a ) P ( x ; a )

exist, x * is called as a regular singular point. Otherwise, x * is said to be an irregular singular point.

Proof

Here a is a singular point of (42) since P ( a ; a ) = 0 . Furthermore, a is regular singular point because

lim x → a ( x − a ) α 1 Q ( x ; a ) P ( x ; a ) = lim x → a ( x − a ) α 1 ( x − a ) α 1 t q + ( 1 − t ) q ( x − a ) α 2 = lim x → a ( x − a ) α 1 t q + ( 1 − t ) q ( x − α ( a ) ) < ∞ ,

where we used additivity rule (12) and

lim x → a ( x − a ) α 2 R ( x ; a ) P ( x ; a ) = lim x → a ( x − a ) α 2 λ 2 ( x − a ) α 2 t q + ( 1 − t ) q ( x − a ) α 2 = lim x → a λ 2 ( x − a ) α 2 t q + ( 1 − t ) q < ∞ .

To construct the solutions of (42) about a , we use the formal power series

u ( x ; α i ( a ) ) ≔ ∑ j = 0 ∞ b j ( x − α i ( a ) ) α j + r , r ∈ R , i = 1 , 2 .

By using the additivity rule (12), we calculate the third term in (42) as follows:

(45) ( x − a ) α 2 u ( x ; α 2 ( a ) ) = ∑ j = 0 ∞ b j ( x − a ) α 2 ( x − α 2 ( a ) ) α j + r = ∑ j = 0 ∞ b j ( x − a ) α j + r + 2 = ∑ j = 2 ∞ b j − 2 ( x − a ) α j + r ,

while the second term in (42) arises by the use of the power rule for derivatives (18) and additivity rule (12)

(46) ( x − a ) α 1 D α u ( x ; α ( a ) ) = ∑ j = 0 ∞ b j [ j + r ] α ( x − a ) α 1 ( x − α ( a ) ) α j + r − 1 = ∑ j = 0 ∞ b j [ j + r ] α ( x − a ) α j + r .

Similarly, the first term in (42) is computed as follows:

(47) ( x − a ) α 2 D α 2 u ( x ; α 2 ( a ) ) = ∑ j = 0 ∞ b j [ j + r ] α [ j + r − 1 ] α ( x − a ) α 2 ( x − α 2 ( a ) ) α j + r − 2 = ∑ j = 0 ∞ b j [ j + r ] α [ j + r − 1 ] α ( x − a ) α j + r .

We substitute (45), (46), and (47) in (42) and obtain

(48) ∑ j = 2 ∞ t q + ( 1 − t ) q b j [ j + r ] α [ j + r − 1 ] α + b j [ j + r ] α + λ 2 b j − 2 ( x − a ) α j + r

(49) + t q + ( 1 − t ) q b 0 [ r ] α [ r − 1 ] α + b 0 [ r ] α ( x − a ) α r

(50) + t q + ( 1 − t ) q b 1 [ r + 1 ] α [ r ] α + b 1 [ r + 1 ] α ( x − a ) α r + 1 = 0 .

The main term in (48) implies that

(51) b j = − λ 2 b j − 2 [ j + r ] α t q + ( 1 − t ) q [ j + r − 1 ] α + 1 = − λ 2 b j − 2 [ j + r ] α 2 .

From the term in (49), we obtain

b 0 [ r ] α t q + ( 1 − t ) q [ r − 1 ] α + 1 = b 0 [ r ] α [ r ] α = b 0 [ r ] α 2 = 0 .

By assuming b 0 = 1 , we conclude that the indical equation [ r ] α 2 = 0 , that is, r 1 = r 2 = 0 . Similarly, from the term (50), one can obtain b 1 [ r + 1 ] α 2 = 0 . Since r = 0 , then b 1 = 0 , and using (51), we conclude that b 2 j + 1 = 0 for all j ∈ N and the even indexed terms can be found as follows:

(52) b 2 j = ( − 1 ) j λ 2 j ( [ 2 ] α [ 4 ] α … [ 2 j ] α ) 2 .

Here, by using the definition of α -numbers (3), we can derive the identity

[ 2 j ] α = [ 2 ] α [ j ] α 2

and by Proposition 4.10 (ii), we have Γ α 2 ( j + 1 ) = [ j ] α 2 ! . Hence, (52) can be rewritten as follows:

b 2 j = ( − 1 ) j λ 2 j [ 2 ] α 2 j ( Γ α 2 ( j + 1 ) ) 2 .

One can check that the function (44) satisfies equation (42).

We apply the ratio test to determine the radius of convergence of the function (44). By using Theorem 4.10 (ii) and the additivity rule (12), we obtain the limit

L = lim j → ∞ b 2 ( j + 1 ) ( x − a ) α 2 ( j + 1 ) b 2 j ( x − a ) α 2 j = lim j → ∞ λ 2 ∣ ( x − α 2 j ( a ) ) ( x − α 2 j + 1 ( a ) ) ∣ [ 2 ] α 2 [ j + 1 ] α 2 = λ 2 x − h 1 − q 2 [ 2 ] α 2 1 1 − t q + ( 1 − t ) q 2 2 = λ 1 − t q + ( 1 − t ) q x − h 1 − q 2 < 1

if ∣ x − h 1 − q ∣ < 1 λ 1 − t q + ( 1 − t ) q . □

Case III. η < 0 : Set η = − λ 2 . Similar to Case II, by Proposition 4.3, w ( τ ; a ) = Exp α ( λ 2 K ( τ − a ) ) solves the equation D α w ( τ ; a ) − λ 2 K w ( τ ; a ) = 0 . In this case, (38) provides

(53) t q + ( 1 − t ) q ( x − a ) α 2 D α 2 u ( x ; α 2 ( a ) ) + ( x − a ) α 1 D α u ( x ; α ( a ) ) − λ 2 ( x − a ) α 2 u ( x ; α 2 ( a ) ) = 0 ,

which is the α -analogue of modified Bessel equation of order zero since it recovers the ordinary modified Bessel equation of order zero

x 2 u ″ + x u ′ − x 2 u = 0 ,

when λ = 1 , a = 0 and ( q , h ) → ( 1 , 0 ) . The following theorem is dedicated to present a solution to modified α -Bessel equation (53), with computations similar to those in Theorem 5.2.

Theorem 5.4

A solution of the modified α -Bessel equation of order zero (53) is presented as follows:

(54) I 0 α ( λ ( x − a ) ) ≔ ∑ j = 0 ∞ λ 2 j ( x − a ) α 2 j [ 2 ] α 2 j ( Γ α 2 ( j + 1 ) ) 2 ,

and converges absolutely if ∣ x − h 1 − q ∣ < 1 ∣ λ ∣ 1 − t q + ( 1 − t ) q . When λ = 1 , we call (54) as the modified α -Bessel function of order zero of the first kind.

Remark 5.5

As a conclusion, α -cylindrical diffusion equation (36) admits the following solution:

z ( x , τ ; a ) = ln x − h 1 − q if η = 0 , J 0 α ( λ ( x − a ) ) Exp α ( − λ 2 K ( τ − a ) ) if η = λ 2 , I 0 α ( λ ( x − a ) ) Exp α ( λ 2 K ( τ − a ) ) if η = − λ 2 ,

where the solution is determined explicitly in terms of α -analogues of exponential, gamma, Bessel, and modified Bessel functions.

6 Conclusion

In this work, our primary objectives were to develop the α -exponential and α -gamma functions. During our investigations, we discovered α -Bessel and modified α -Bessel functions of order zero of the first kind. The second linearly independent solutions of α -Bessel equation (42) and modified α -Bessel equation (53) are open problems, which will be analyzed in an ongoing project. Furthermore, we will continue to study α -analogues of other special functions as a future work.

Funding information: This research received no specific grant from any funding agency in the public, commercial, or nonprofit sectors.
Author contributions: The main ideas and overall structure, the mathematical analysis, and the development of the proofs of the article were jointly developed by all authors. All authors discussed the results and approved the final version of the manuscript. All authors have accepted responsibility for entire content of this article and approved its submission.
Conflict of interest: The authors declare that they have no conflict of interest.
Ethical approval: Not applicable. This study does not involve human participants or animal subjects.
Data availability statement: No datasets were generated or analyzed during the current study.

References

[1] M. Cichoń, B. Silindir, A. Yantir, and S. Gergün, Generalized polynomials and their unification and extension to discrete calculus, Symmetry 15 (2023), no. 9, 1677, 1–26, DOI: https://doi.org/10.3390/sym15091677. 10.3390/sym15091677Search in Google Scholar

[2] J. Čermák and L. Nechvátal, On (q,h) -analogue of fractional calculus, J. Nonlinear Math. Phys. 17 (2010), no. 1, 51–68, DOI: https://doi.org/10.1142/S1402925110000593. 10.1142/S1402925110000593Search in Google Scholar

[3] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser, Boston, 2001. 10.1007/978-1-4612-0201-1Search in Google Scholar

[4] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56, DOI: https://doi.org/10.1007/BF0332315310.1007/BF03323153Search in Google Scholar

[5] V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, Springer, New York, 1996. 10.1007/978-1-4757-2449-3Search in Google Scholar

[6] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. 10.1007/978-0-8176-8230-9Search in Google Scholar

[7] B. D. Haile and L. M. Hall, Polynomial and series solutions of dynamic equations on time scales, Dynam. Systems Appl. 12 (2003), 149–157. Search in Google Scholar

[8] D. R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, Panam. Math. J. 23 (2002), 17–27. Search in Google Scholar

[9] D. Mozyrska and D. F. M. Torres, Diamond-alpha polynomial series on time scales, Int. J. Math. Stat. 5 (2009), no. A09, 92–101. Search in Google Scholar

[10] B. Silindir, S. Gergün, and A. Yantir, Analysis on α-time scales and its applications to Cauchy-Euler equation, Appl. Math. Inf. Sci. 18 (2024), no. 5, 1051–1074, DOI: https://doi.org/10.18576/amis/180512. 10.18576/amis/180512Search in Google Scholar

[11] B. Silindir and A. Yantir, Generalized quantum exponential function and its applications, Filomat 33 (2019), no. 15, 4907–4922, DOI: https://doi.org/10.2298/FIL1915907S10.2298/FIL1915907SSearch in Google Scholar

[12] A. Yantir, B. Silindir, and Z. Tuncer, Bessel equation and Bessel function on T(q,h), Turkish J. Math. 46 (2022), no. 8, 3300–3322, DOI: https://doi.org/10.55730/1300-0098.3334. 10.55730/1300-0098.3334Search in Google Scholar

[13] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002. 10.1007/978-1-4613-0071-7Search in Google Scholar

[14] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1976. Search in Google Scholar

[15] W. F. Trench, Introduction to Real Analysis, Faculty Authored and Edited Books & CDs. vol. 7, 2013, https://digitalcommons.trinity.edu/mono/7/. Search in Google Scholar

[16] R. C. Buck, Advanced Calculus, Waveland Press, Inc., Illinois, 2003. Search in Google Scholar

[17] B. Silindir and A. Yantir, Gaussas binomial formula and additive property of exponential functions on T(q,h), Filomat 35 (2021), no. 11, 3855–3877, DOI: https://doi.org/10.2298/FIL2111855S. 10.2298/FIL2111855SSearch in Google Scholar

[18] A. De Sole and V. G. Kac, On integral representations of q-gamma and q-beta functions, Atti Accad. Naz. Lincei Mat. Appl. 16 (2005), no. 1, 11–29. Search in Google Scholar

[19] S. Gergün, B. Silindir, and A. Yantir, Power function and binomial series on T(q,h), Appl. Math. Sci. Eng. 31 (2023), no.1, 2168657:1–18, DOI: https://doi.org/10.1080/27690911.2023.2168657. 10.1080/27690911.2023.2168657Search in Google Scholar

[20] D. Gronau, Normal solutions of difference equations, Euleras functions ans spirals, Aequationes Math. 68 (2004), 230–247, DOI: https://doi.org/10.1007/s00010-004-2765-3. 10.1007/s00010-004-2765-3Search in Google Scholar

[21] M. Bohner and B. Karpuz, The gamma function on time scales, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 (2013), 507–522. Search in Google Scholar

[22] H. M. Srivastava, K.-Y. Kung, S.-F. Lee, and S.-D. Lin, Analytic solutions of the cylindrical heat equation with a heat source, Russ. J. Math. Phys. 28 (2021), 545–552, DOI: https://doi.org/10.1134/S1061920821040129. 10.1134/S1061920821040129Search in Google Scholar

[23] T. J. Cuchta, Discrete Analogues of Some Classical Special Functions, PhD. thesis, Missouri University of Science and Technology, Missouri, 2015. Search in Google Scholar

[24] M. Bohner and T. Cuchta, The Bessel difference equation, Proc. Amer. Math. Soc. 145 (2017), 1567–1580, DOI: https://doi.org/10.1090/proc/13416. 10.1090/proc/13416Search in Google Scholar

[25] W. Hahn, Beitrage zur theorie der Heineschen Reichen, Math. Nachr. 2 (1949), 340–379 (in German). 10.1002/mana.19490020604Search in Google Scholar

[26] F. H. Jackson, The application of basic numbers to Bessel’s and Legendre’s functions, Proc. Lond. Math. Soc. 2 (1905), 192–220, DOI: https://doi.org/10.1112/plms/s2-2.1.192. 10.1112/plms/s2-2.1.192Search in Google Scholar

Received: 2024-11-01

Accepted: 2025-04-09

Published Online: 2025-06-10

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

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0131

Keywords for this article

exponential function; gamma function; Bessel equations; Bessel functions; cylindrical diffusion equation

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