No, and ask your GM respectively. Synthesist doesn't have a lot of detail and Paizo is not likely to provide any updates/FAQ for it since it's theoretically not allowed for the unchained summoner, and that's the one they plan to use going forward.
[b]Summoner: Can a synthesist (page 80) make attacks from his own body (such as manufactured weapons, unarmed strikes, or natural attacks) and attacks from the fused eidolon in the same round? [b]
Yes, but the fused character's natural attacks are still subject to the Maximum Attacks entry in the table for an eidolon of his level. For example, a 1st-level synthesist is limited to 3 natural attacks per round, whether those natural attacks are from the eidolon, the synthesist, or a combination of the two.
If the synthesist wants to use his eidolon's natural attacks and use his own manufactured weapons or natural weapons in the same round, his eidolon needs to have enough limbs to account for all of these attacks. For example, a gnome synthesist fused with a two-armed biped eidolon has two arms it can use to make attacks; if the synthesist wants to make claw attacks with his eidolon's claws and also make weapon attacks (such as with a dagger or staff), he needs to give his eidolon additional arms evolutions to hold those weapons (as an extension of the summoner's own limbs)--a two-armed eidolon can't make two claw attacks and also make a dagger attack or staff attack in the same round.
Remember that the synthesist is still subject to the rules of combining manufactured weapon attacks and natural weapon attacks in the same round (in that the natural weapons are always considered secondary and therefore have a -5 attack penalty).
Remember also that the summoner is wearing the eidolon like a biological, all-encompassing "suit," and the eidolon's shape limits what the summoner can do. If the eidolon doesn't have arms, the summoner can't use his own arms to manipulate objects, make attacks, cast somatic spells, or anything else requiring arms--while fused, the summoner's limbs are trapped within the armless eidolon-suit, and he isn't able to use them to manipulate things. The summoner isn't able to extend his own body parts outside of the eidolon-suit; if he wants to be able to manipulate things with arms, the eidolon needs arms (though tentacles are sufficient for simple tasks).
(Note: It is a matter of flavor and player's preference whether the synthesist floats immobile within the eidolon-suit and its limbs move at his mental command, if the synthesist moves his own arms and the eidolon-suit's arms echo this movement, or if the eidolon-suit is more form-fitting and the flesh-enveloping limbs move in direct response to the synthesist's own movements.)
Note: This clarifies an earlier FAQ error where the summoner's weapon attacks counted toward the number of attacks on the table.
In "The One Synthesist Thread to Rule Them All" thread it was clarified that a synthesis could not cast spells requiring somatic components while fused unless the eidolon had arms.