research & reviews discrete mathematical structures vol 3 issue 3

Discrete Mathematical Structures

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STM JOURNALS

1. Fractional Calculus of the R-Series Mohd. Farman Ali, Manoj Sharma 1

2. R-L F Integral and Triple Dirichlet Average of the R-Series Mohd. Farman Ali 6

3. Dirichlet Average of New Generalized M-series and Fractional CalculusManoj Sharma 13

4. A Brief Review on Algorithms for Finding Shortest Path of Knapsack ProblemSwadha Mishra 17

5. Research and Industrial Insight: Discrete Mathematics 20

ContentsResearch & Reviews: Discrete Mathematical Structures

RRDMS (2016) 1-5 © STM Journals 2016. All Rights Reserved Page 1

Research & Reviews: Discrete Mathematical Structures ISSN: 2394-1979(online)

Volume 3, Issue 3

www.stmjournals.com

Fractional Calculus of the R-Series

Mohd. Farman Ali1,*, Manoj Sharma

2

1Department of Mathematics, Madhav University, Sirohi, Rajasthan, India

2Department of Mathematics, Rustamji Institute of Technology, BSF Academy, Tekanpur, Gwalior,

Madhya Pradesh, India

Abstract The present paper creates a special function called as R-series. This is a special case of H-

function given by Inayat Hussain. The Hypergeometric function, Mainardi function and M-

series follow R-series and these functions have recently found essential applications in solving

problems in physics, biology, bio-science, engineering and applied science etc.

Mathematics Subject Classification—26A33, 33C60, 44A15

Keywords: Fractional calculus operators, R−series, Mellin-Barnes integral, special functions

INTRODUCTION TO THE H-FUNCTION The H- function of Inayat Hussain, is a generalization of the familiar H-function of Fox, defined in

terms of Mellin-Barnes contour integral [1], as

𝐻𝑝,𝑞𝑚,𝑛 [𝑧

(𝛽𝑗,𝐵𝑗)1,𝑚 ,(𝛽𝑗,𝐵𝑗; 𝑏𝑗 )𝑚+1,𝑞

(𝛼𝑗,𝐴𝑗;𝛼𝑗)1,𝑛 ,(𝛼𝑗,𝐴𝑗;𝛼𝑗)𝑛+1,𝑝 ] =

1

2𝜋𝑖∫ 𝜃(𝑠)𝑧𝑠𝑑𝑠

+𝑖∞

−𝑖∞ (1)

Where the integrand (or Mellin transform of the H-function)

𝜃(𝑠) =∏ Γ(𝛽𝑗−𝐵𝑗𝑠) ∏ [Γ(1=𝛼𝑗+𝐴𝑗𝑠)]

𝛼𝑗𝑛𝑗=1

𝑚𝑗=1

∏ [Γ(1=𝛽𝑗+𝐵𝑗𝑠)]𝑏𝑗𝑛

𝑗=𝑚+1 ∏ Γ𝑝𝑗=𝑛+1 (𝛼𝑗−𝐴𝑗𝑠)

(2)

Contains fractional powers of some of the involved Γ −functions. Here 𝛼𝑗(1𝑒𝑟𝑒𝑝) and 𝛽𝑗(1𝑛𝑑 𝑞) are

complex parameters; 𝐴𝑗 > 0 ( 10𝑟𝑒𝑝 ), 𝐵𝑗 > 0(1… … … 𝑞) ; and exponents; 𝑎𝑗(𝑗 = 1 … … … 𝑛) and

𝑏𝑗(𝑗 = 1 … … … 𝑞) can take noninteger values. Evidently, when all the exponent 𝑎𝑗 and 𝑏𝑗 take integer

values only, the H-function reduces to the familiar H-function of Fox, [1–3]. The sufficient conditions

for the absolute convergence of the contour integral (1), as given by Buschman and Srivastava, are as

follows [3]:

Ω = 𝐵𝑗 𝑚𝑗=1 + 𝑎𝑗𝐴𝑗 −𝑛

𝑗=1 𝑏𝑗𝐵𝑗 𝑝𝑗=𝑚+1 − 𝐴𝑗

𝑞𝑗=𝑛+1 > 0 and arg(𝑧) <

1

2𝜋 Ω

THE R-SERIES The R- series is

𝑅𝑞𝛼,𝛽

𝑝0 (𝑎1 . . . .𝑎𝑝; , 𝑏1 . . . . 𝑏𝑞; 𝑧

0) = 𝑅𝑞𝛼,𝛽

𝑝0 (𝑧)

𝑅𝑞𝛼,𝛽

𝑝0 (𝑧) =

(𝑎1)𝑘 . . . . .(𝑎𝑝)𝑘

(𝑏1)𝑘 . . . . .(𝑏𝑞)𝑘

∞𝑘=𝑜

𝑧𝑘

Γ(𝛼𝑘+𝛽)𝑘! (3)

Here, 𝑝 upper parameters 𝑎1,𝑎2, . . . . 𝑎𝑝 and 𝑞 lower parameters 𝑏1, 𝑏2, . . . .𝑏𝑞 , 𝛼𝜖𝐶 , 𝑅(𝛼) >

0,𝑚 > 0 and (𝑎𝑗)𝑘 (𝑏𝑗)𝑘



Volume 3, Issue 3

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R-L F Integral and Triple Dirichlet Average of the

R-Series

Mohd. Farman Ali Department of Mathematics, Madhav University, Sirohi, Rajasthan, India

Abstract In this article, we establish the relation between some results of triple Dirichlet average of the

R-series and fractional operators. We use a new special function called as R-series, which is a

special case of H-function given by Inayat Hussain. In this article, the solution is obtained in

compact form of triple Dirichlet average of R-series as well as conversion into single

Dirichlet average of R-series, using fractional integral.

Keywords: Dirichlet averages, special functions, R-series and Riemann-Liouville fractional

integral

Mathematics Subject Classification: 2000: Primary: 33E12, 26A33; Secondary: 33C20,

33C65.

INTRODUCTION The Dirichlet average of a function is a certain kind integral average with respect to Dirichlet

measure. The concept of Dirichlet average was introduced by Carlson in 1977. Carlson has defined

Dirichlet averages of functions, which represent certain types of integral average with respect to

Dirichlet measure [1–4]. He showed that various important special functions could be derived as

Dirichlet averages for the ordinary simple functions like 𝑥𝑡,𝑒𝑥 etc. He has also pointed out that the

hidden symmetry of all special functions, which provided their various transformations can be

obtained by averaging 𝑥𝑛,𝑒𝑥 etc. [5, 6]. Thus, he established a unique process towards the unification

of special functions by averaging a limited number of ordinary functions [7].

Gupta and Agarwal found that averaging process is not altogether new but directly connected with the

old theory of fractional derivative [8, 9]. Carlson overlooked this connection whereas he has applied

fractional derivative in so many cases during his entire work. Deora and Banerji have found the

double Dirichlet average of ex by using fractional derivatives and they have also found the triple

Dirichlet average of xt by using fractional derivatives [10, 11].

Sharma and Jain obtained double Dirichlet average of trigonometry function cos 𝑥 using fractional

derivative and they have also found the triple Dirichlet average of ex by using fractional calculus

[12–15].

Recently, Kilbas and Kattuveetti established a correlation among Dirichlet averages of the generalized

Mittag-Leffler function with Riemann-Liouville fractional integrals and of the hyper-geometric

functions of many variables [16].

DEFINITIONS AND PRELIMINARIES Some definitions are necessary in the preparation of this paper.

Standard Simplex in 𝑹𝒌, 𝒌 ≥ 𝟏:

The standard simplex in 𝑅𝑘, 𝑘 ≥ 1 by [1].

𝐸 = 𝐸𝑘 = {𝑆(𝑢1,𝑢2, … 𝑢𝑘) ∶ 𝑢1 ≥ 0, … 𝑢𝑘 ≥ 0, 𝑢1 + 𝑢2 + ⋯ + 𝑢𝑘 ≤ 1}



Volume 3, Issue 3

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Dirichlet Average of New Generalized

M-series and Fractional Calculus

Manoj Sharma Department of Mathematics, Rustamji Institute of Technology, BSF Academy, Tekanpur,

Gwalior, Madhya Pradesh, India

Abstract We know that every analytic function can be measured as a Dirichlet average and connected

with fractional calculus. In this note, we set up a relation between Dirichlet average of new

generalized M-series, and fractional derivative. Fractional derivative is a derivative of

arbitrary order i.e. may be real, complex, integer or fractional order.

Mathematics Subject Classification: 26A33, 33A30, 33A25 and 83C99.

Keywords: Dirichlet average new generalized M-series, fractional derivative, fractional

calculus operators

INTRODUCTION Carlson has defined Dirichlet average of functions which represents certain types of integral average

with respect to Dirichlet measure [1–5]. He showed that various important special functions can be

derived as Dirichlet averages for the ordinary simple functions like𝑥𝑡,𝑒𝑥 etc. He has also pointed out

that the hidden symmetry of all special functions which provided their various transformations can be

obtained by averaging 𝑥𝑛,𝑒𝑥 etc. [6, 7]. Thus he established a unique process towards the unification

of special functions by averaging a limited number of ordinary functions. Almost all known special

functions and their well-known properties have been derived by this process.

In this paper, the Dirichlet average of new generalized M-series has been obtained.

DEFINITIONS We give below some of the definitions which are necessary in the preparation of this paper:

Standard Simplex in 𝑹𝒏, 𝒏 ≥ 𝟏:

We denote the standard simplex in 𝑅𝑛, 𝑛 ≥ 1 by Carlson [1].

𝐸 = 𝐸𝑛 = {𝑆(𝑢1,𝑢2, … … . . 𝑢𝑛) ∶ 𝑢1 ≥ 0, … … … . 𝑢𝑛 ≥ 0, 𝑢1 + 𝑢2 + ⋯ … … + 𝑢𝑛 ≤ 1} (1)

Dirichlet Measure

Let 𝑏 ∈ 𝐶𝑘 , 𝑘 ≥ 2 and let 𝐸 = 𝐸𝑘−1 be the standard simplex in 𝑅𝑘−1. The complex measure 𝜇𝑏 is

defined by 𝐸[1].

𝑑𝜇𝑏(𝑢) =1

𝐵(𝑏)𝑢1

𝑏1−1… … … … … . 𝑢𝑘−1

𝑏𝑘−1−1(1 − 𝑢1 − ⋯ … … … − 𝑢𝑘−1)𝑏𝑘−1𝑑𝑢1 … … … … . 𝑑𝑢𝑘−1 (2)

It will be called a Dirichlet measure.

Here,

𝐵(𝑏) = 𝐵(𝑏1, … … … . 𝑏𝑘) =Γ(𝑏1) … … … … … . . Γ(𝑏𝑘)

Γ(𝑏1 + ⋯ … … . . +𝑏𝑘),

𝐶> = {𝑧 ∈ 𝑧: 𝑧 ≠ 0, |𝑝ℎ 𝑧| < 𝜋2⁄ },

Open right half plane and 𝐶>k is the 𝑘𝑡ℎ Cartesian power of 𝐶>.



Volume 3, Issue 3

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A Brief Review on Algorithms for Finding Shortest Path

of Knapsack Problem

Swadha Mishra* Department of Computer Applications, Invertis University, Bareilly, Uttar Pradesh, India

Abstract The gathering knapsack and knapsack problems are summed up to briefest way issue in a

class of graphs. An effective calculation is used for finding briefest ways that bend lengths are

non-negative. A more effective calculation is portrayed for the non-cyclic which incorporates

the knapsack issue.

Keywords: knapsack problem, rucksack problem

INTRODUCTION The knapsack problem or rucksack problem is

an issue in combinatorial enhancement: Given

an arrangement of things, each with a weight

and an esteem, decide the number of

everything to incorporate into an accumulation

so that the aggregate weight is not exactly or

equivalent to a given point of confinement and

the aggregate esteem is as expansive as could

reasonably be expected.

Group knapsack problem has been given to:

Minimise ∑ 𝑐𝑛𝑗=1 jjxj (1)

Subject to ∑ 𝑐𝑛𝑗=1 jgj=g0 (2)

Where, x1,…, xn non-negative integers.

g0,…, gn are the subset of the elements of a

finite additive abelian group H and c1,…,cn are

non-negative reals.

This algorithm for solving this problem has

been described by Gomory [1], Shapiro [2, 3],

Hu [4] and others. It can be formulated as a

shortest path problem in the following way.

Let G1 be the graph with node H and arc of the

form (h, h+gj) h an arbitrary element of H and

j=1,…, n. The length of such an arc is cj. Let P

be the from 0 to g0 in G1 then if xj is the

number of arcs of the form (h, h+gj) in P then

(x1,…, xn) is a solution to Eq. (2) and the

length of P is Eq. (1). Conversely, if (xn,…, xn)

satisfied Eq. (2), then one may construct paths

from 0 to g0. Now a new algorithm is given in

this paper to solve this problem [5].

The name knapsack problem applies to:

Maximize ∑ 𝑐𝑛𝑗=0 jjxj (3)

Subject to ∑ 𝑤𝑛𝑗=0 jxj=W, (4)

Where, x0, xi,…, xn non-negative integers.

Where, c0=0, c1,…, cn are positive reals, w0=1

and w1,…, wn, W are positive integers.

One can formulate a knapsack problem as a

longest path problem defining the graph G2

with nodes 0, 1,…, W and arcs of the form (w,

w+wj) of length cj. The knapsack problem is

then equivalent to that of finding a longest

path from 0 to W [6].

ALGORITHM The graph G1 and G2 of the previous section

are examples of a class of graphs which for the

purposes of this paper, we call knapsack

graphs.

Definition

A graph G with nodes N and arcs A is a

knapsack graph if,

(1) The arcs A can be partitioned into n

disjoint sets A1,…,AN;

(2) The length of each arc belonging to Aj is lj;

(3) Let P=(i0, i1,...,ip) be a path between an

arbitrary pair of nodes i0, ip.

Suppose that (it-1, it)∈Amt, for t=1,…., p.

Then for any re-ordering, n1,…., np of the

indices m1,…, mp, there exist a path

Q=(j0,j1,…, jp), where j0=i0, jp=ip and (jt-1=jt)

∈Ant for t=1,…., p.

For shortest path problems with non-negative



Volume 3, Issue 3

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Research and Industrial Insight: Discrete Mathematics

Math’s Maze Runner

Mazes are in vogue right now, from NBO's

West world, to the arrival of the British faction

TV arrangement, The Crystal Maze. Be that as

it may, labyrinths have been around for

centuries and a standout amongst the most

acclaimed labyrinths, the Labyrinth home of

the Minotaur, assumes a featuring part in

Greek mythology.

The most part acknowledged that a maze

contains just a single way, regularly spiraling

around and collapsing back on it, in constantly

diminishing circles, though a labyrinth

contains expanding ways, giving the voyager

decisions and the potential for getting,

exceptionally lost.

Design a maze is very tough task and for that

human should be rewarded. Many algorithms

for maze have been created by computer

scientists and mathematicians. These

algorithms based on two principles: one which

begin with a solitary, limited space and

afterward sub-separate it with dividers (and

entryways) to deliver ever littler sub-spaces;

and others which begin with a world brimming

with detached rooms and after that devastate

dividers to make ways/courses between them.

Escape Plan

There are many techniques by which you can

run away from maze but firstly you need to the

details about the maze from which you want to

escape. Most strategies work for "basic"

labyrinths, that is, ones with no tricky alternate

routes by means of scaffolds or "entry circles"

– round ways that lead back to where they

began.

Along these lines, accepting it is a

straightforward labyrinth, the strategy that

many individuals know is "follow the wall”.

Basically, you put one hand on a mass of the

labyrinth (it doesn't make a difference which

hand the length of you are reliable) and after

that continue strolling, keeping up contact

between your hand and the divider. In the end,

you will get out. This is on account of on the

off chance that you envision getting the mass

of a labyrinth and extending its edge to

evacuate any corners, you will in the long run

frame something circle-like, a portion of

which must shape part of the labyrinth's

external limit. This strategy for escape may

not work, be that as it may, if the begin or

complete areas are in the labyrinth's middle.

Be that as it may, a few labyrinths are

purposely intended to disappoint, for example,

the Escot Gardens' beech fence labyrinth in

Devon, which contains no less than five

extensions, thus a long way from

"straightforward".

There is another method by which maze

escape will be easy i.e., Tremaux’s Algorithm,

it works in all the cases.

Envision that, as Hansel and Gretel in the pixie

story, you can leave a trail of "breadcrumbs"

behind you as you explore your way through

the labyrinth and afterward recall these

guidelines: in the event that you touch base at

an intersection you have not beforehand

experienced (there will be no scraps as of now

on the trail ahead), then haphazardly select an

approach. On the off chance that that leads you

to an intersection where one way is different to

you yet the other is not, then select the

unexplored way. What's more, if picking

between an on more than one occasion utilized

way, pick the way utilized once, then leave

another, second trail behind you. The cardinal

administer is never, ever select a way as of

now containing two trails. This technique is

ensured, in the end, to get you out of any

labyrinth.

Mazes in everyday life

Now you think how is this maze thing useful

for us? Maze is interesting and adventurous

but not for our daily day-to-day life when we

are on work or we are doing something

important.

Bill Hillier, a theorist in 1980s, find that most

of the housing estates that have a layout like

maze. This reise q question that we can

measure the maze-iness of a house?

Discrete Mathematical Structures

(RRDMS)May–August 2016

Research & Reviews:

ISSN 2394-1979 (Online)

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