There is a saying that a child born in Moola, Ashlesya, Jyeshta, Maga is inauspicious. Is this properly explained in any religious books? This question is pursuant to astrology.
What you read/heard is only the partial truth.There are two more nakshtras which are considered equally inauspicious as per the "Gandamoola" concept viz-Aswini and Revati.
There are 27 nakshtras in all in Jyotish.Of them 6 nakshtras, 3 ruled by Ketu(viz-Aswini,Magha & Mula) & 3 ruled by Budha(viz-Revati,Ashlesa &Jyestha) are called "Gandamoola nakshtras".Birth in them are considered inauspicious.Infact,not only Moon,other Grahas are also supposed to give malefic results if placed in the "Gandantas".
Gandanta is the junction points in the zodiac that a particularly connected with times of soul growth. This is the position of the zodiac where the where the solar and lunar Zodiac meet. These points are at the junctions of Pisces (Revati)- Aries (Ashwini), Cancer (Ashesha)- Leo (Magha), Scorpio (Jyeshta)- Sagittarius (Mula). They are especially powerful in giving Karmic effects.
There are 27 Nakshatras in Vedic Astrology; in the modern era most of the astrologers count 28 Nakshatras including Abhijit Nakshatra. Those Nakshatras which are ruled by Ketu and Mercury are known as Gandmool Nakshatra. These are basically 6 nakshatras namely Ashwini, Ashlesha, Magha, Jyestha, Moola and Revati.
One of the authentic Jyotisha text, The Brihat Parashara Hora Shastra ,mentions that birth in these nakshtras are considered inauspicious.
1-4. The Venerable Sage said. O Maitreya! Now I will describe to you the circumstances, in which the births are inauspicious in spite of Lagn and the Grahas being well disposed. Although Lagn may be well disposed, births will be inauspicious, if they take place on Amavasya (last day of the Krishna Paksha), on Chaturdasi (14th Tithi), in Krishna Paksha (dark half of the month), in Bhadra Karan, in the Nakshatr of the brother, in the Nakshatras of father and mother, at the time of entry of Surya in a Rashi, the time of Pata, at the time of solar and lunar eclipses, at the time of Vyati Pata, in Gandantas of all the three kinds, in Yamaghant, Tithikshaya, in Dagdha Yog etc.
Remedies include chanting the Janma Nakshatra mantras etc.
I do not know where you heard this saying but it is far from true.
The birth star is the star-group the Moon occupies at the time of the birth.
Ashlesha and Jyeshtha are rules by Budha (Mercury). Budha is friendly to Chandra (Moon), although their nature are quite different. Budha is the intellect and Chandra is the emotion. Hence, there may be some conflict at times.
Also, Mula and Magha are ruled by Ketu. Now, one might argue that Ketu is a spiritual planet and may give hardship in the material plane. Although, in that case, same logic should be applied to Rahu or Shani.
To sum it up, even if there is any inauspicious effect at all, it is a small part of a very big picture. When looking at all the dimensions of a chart, these combinations lose their relative value.
There is an Alwar born in one of this nakshatra. Place Thirumandangudi Month Maargazhi Star Kettai (Jyeshta)
Similarly we can find more examples of spiritual Giants born in these nakshatras.
So inauspicious has no meaning for devotees of God. We can always strive to make the children devoted to Vasudeva and that will take care of all such concerns if there are any.