In this article, our main purpose is to establish a new extension of Mittag-Leffler function by using the known extended beta function $\mathbf{B}_{\omega}(a,b;p)$ introduced in [1]. It led to a novel extension of the applicability of Mittag-Leffler function that introduced them as distributions defined for a specific set of functions. We also, investigate some of its important properties, namely recursion relation, Mellin transform and differential formulas.

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.

In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.

The main aim of this paper is to establish two generalized integral formulas involving the extended Mittag–Leffler function based on the well known Lavoie and Trottier integral formula and the obtain results are express in term of extended Wright-type function. Also, we establish certain special cases of our main result.

Assessment of the genetic distinctiveness of a cultivar through morphological descriptors is an important tool for both the registration and the protection. New mango genotypes have been improved using valuable diverse germplasm resources to ensure food security. DNA fingerprinting based simple sequence repeats (SSR)-markers have been the most broadly used, effective and accurate in evaluation of genetic characterization of a cultivar. Molecular breeding is an effective source of genetic gain after improvement of fruit trees using marker assisted genomic selection. Mango (Mangifera indica L.) is an allotetraploid (2n = 4X= 40) drupe fruit and has high nutritional value belongs to genus Mangifera and family Anacardiaceae. Mango cultivars are used with worldwide acceptance to pharmacological, ethnomedical, and phytochemical industries. This study investigated the molecular evaluation of a new mango cultivar ‘Azeem Chaunsa’ using a set of the most effective 50 hyper-variable polymorphic SSR markers. Highly specific DNA fingerprints were identified in the genome of this mango cultivar, ‘Azeem Chaunsa’ compared with three standard cultivars such as Sindhri, Samar Bahisht (S.B) Chaunsa and Sufaid Chaunsa. Our results showed that SSR markers could efficiently assess genetic diversity in mango. An agglomerative hierarchical clustering method was used to construct dendrogram based on the ‘Unweighted Pair- Group Method with Arithmetic Mean’ (UPGMA). The genetic similarity coefficients were recorded between the mangos cultivars ranged from 0.49 to 0.67. Cultivar identification (CID) evaluates association among standard cultivars and Azeem Chaunsa and further concludes significant variations. CID results concluded that cultivar ‘Azeem Chaunsa’ varied significantly from the check cultivar, Sindhri (46.2%), S.B Chaunsa (45%) and Sufaid Chaunsa (46.7%). The results obtained in this study will orient cultivar identification strategies for a successful future mango breeding programmes in the context of climate change.

Keywords: Agrobacterium 1; Xenorhabdus 2; NR-AMP 3 T-DNA keyword 4; TI-plasmid 5; Intact/Cured/T-DNA Deleted 6 Sensitive/Resistant 7; EMA_PF2 8; HPLC 9